COMPUTATIONS OF
PHASE CHANGE
by
Damir Juric
A dissertation submitted in partial fulllment
of the requirements for the degree of
Doctor of Philosophy
(Mechanical Engineering)
in The University of Michigan
1996
Doctoral Committee:
Associate Professor Gretar Tryggvason, Chairperson
Assistant Professor Steven L. Ceccio
Associate Professor Werner J. A. Dahm
Professor Herman Merte, Jr.
Dr. David Jacqmin, Research Scientist, NASA LeRC
c
Damir Juric 1996
All Rights Reserved
To Minette and Alana
ii
ACKNOWLEDGEMENTS
I would like to express my deepest love and appreciation to my wife, Minette, for
her enduring love and support, to our daughter, Alana, who brings joy to our hearts
every day and to my parents, Slavko and Ljerka, who deserve all the credit for this
work.
I would like to extend my utmost thanks to my advisor, Professor Gretar Tryggvason, who, through his patience and support, has made my Ph.D. experience at
the University of Michigan thoroughly enjoyable in every way. I am especially grateful for his tremendous guidance and advice as a mentor. I would also like to thank
the members of my dissertation committee, Professors Steven Ceccio, Werner Dahm
and Herman Merte for graciously providing thoughtful advice and encouragement
throughout my work.
I sincerely thank Dr. David Jacqmin of the NASA Lewis Research Center for
his support and constructive interactions as a member of my dissertation committee
and particularly as host during my summer visits to NASA Lewis.
My special gratitude goes to Professor Vedat Arpaci whose deep passion and love
for his profession will always inspire my work.
I would like to acknowledge insightful discussions with Drs. Bruce Murray, Sam
Coriell and Georey McFadden of the National Institute of Standards and Technology
and Professor Charles Peskin of the Courant Institute of New York University.
I thank all of my family and friends who have given me their unselsh encouriii
agement and generous help. Without their collective support this work would not
have been possible.
Financial support for this research was provided by NASA graduate student fellowship NGT-51070 and in part by an Amoco fellowship and NSF Grants CTS913214 and CTS-9503208. Some of the preliminary computations for this thesis
were performed on the supercomputers of the NASA Lewis Research Center.
iv
TABLE OF CONTENTS
DEDICATION : : : : : : : :
ACKNOWLEDGEMENTS
LIST OF FIGURES : : : : :
NOMENCLATURE : : : : :
ABSTRACT : : : : : : : : :
CHAPTER
ii
: : iii
: : vii
:: x
: :xviii
::::::::::::::::::::::::::
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::::::::::::::::::::::::
::::::::::::::::::::::::
::::::::::::::::::::::::
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The State of the Art . . . . . . . . . . . . . . . . . . . . . . .
1.3 Contributions Made in this Work . . . . . . . . . . . . . . . .
1
3
5
II. PURE MATERIAL SOLIDIFICATION . . . . . . . . . . . . .
8
2.1 Introduction . . . . . . . . . . . . .
2.2 Mathematical Formulation . . . . .
2.3 Numerical Method . . . . . . . . . .
2.3.1 Discretization . . . . . . .
2.3.2 Solution Procedure . . . .
2.3.3 Modied Newton Iteration
2.4 Results and Discussion . . . . . . . .
2.4.1 Stable Solidication . . . .
2.4.2 Unstable Solidication . .
2.5 Conclusions . . . . . . . . . . . . . .
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8
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29
42
III. ALLOY SOLIDIFICATION . . . . . . . . . . . . . . . . . . . . . 64
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . 71
3.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . 75
v
3.3.1 Discretization . . .
3.3.2 Solution Procedure
3.4 Results and Discussion . . . .
3.5 Conclusions . . . . . . . . . .
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76
77
78
85
IV. LIQUID-VAPOR PHASE CHANGE WITH FLUID FLOW 94
4.1 Introduction . . . . . . . . . . . . . . . . . .
4.2 Mathematical Formulation . . . . . . . . . .
4.3 Numerical Method . . . . . . . . . . . . . . .
4.3.1 Discretization . . . . . . . . . . . .
4.3.2 Solution Procedure . . . . . . . . .
4.4 Results and Discussion . . . . . . . . . . . . .
4.4.1 Comparison with an Exact Solution
4.4.2 Film Boiling . . . . . . . . . . . . .
4.4.3 Rapid Evaporation . . . . . . . . .
4.5 Conclusions . . . . . . . . . . . . . . . . . . .
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94
98
105
105
108
110
110
112
116
119
V. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.1 Present Work . . . . . . . . . . . . .
5.2 Recommendations for Future Work .
5.2.1 Numerical Enhancements .
5.2.2 Applications . . . . . . . .
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136
138
138
140
APPENDIX : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 145
BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 152
vi
LIST OF FIGURES
Figure
2.1
Stable two-dimensional solidication by a line heat sink - a grid
resolution test and convergence study. . . . . . . . . . . . . . . . . . 45
2.2
Eect of varying the Stefan number. . . . . . . . . . . . . . . . . . . 46
2.3
Eect of varying material properties. . . . . . . . . . . . . . . . . . 47
2.4
Temperature proles. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5
Growth of the liquid/solid interface. . . . . . . . . . . . . . . . . . . 49
2.6
Topology change in stable solidication. . . . . . . . . . . . . . . . . 50
2.7
A grid resolution study for dendritic solidication in an insulated
cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.8
Solid fraction and maximum radius vs. time. . . . . . . . . . . . . . 52
2.9
A ve-fold symmetric interface grows with no grid induced anisotropy. 53
2.10
Simulation of the critical nucleation radius. . . . . . . . . . . . . . . 54
2.11
Eect of varying the liquid to solid thermal conductivity ratio. . . . 55
2.12
Eect of varying the liquid to solid volumetric specic heat ratio. . 56
2.13
Solid fraction vs. time compared with maximum theoretical values.
2.14
Six-fold anisotropy in the surface tension results in growth along six
primary directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.15
A well developed side branch structure resulting from lower surface
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vii
57
2.16
Primary dendrite arm length vs. time. . . . . . . . . . . . . . . . . 60
2.17
Demonstration of topology change with six-fold anisotropy in the
surface tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.18
Planar interface breakdown in dendritic solidication. . . . . . . . . 62
2.19
Growth of a two-dimensional succinonitrile dendrite. . . . . . . . . . 63
3.1
A grid resolution study for alloy solidication. . . . . . . . . . . . . 87
3.2
The interface amplitude vs. time compared with linear theory. . . . 88
3.3
The convergence behavior. . . . . . . . . . . . . . . . . . . . . . . . 89
3.4
Computation of complex microstructure. . . . . . . . . . . . . . . . 90
3.5
The eect of latent heat release at the interface on the heat ow. . . 91
3.6
The eect of capillary anisotropy. . . . . . . . . . . . . . . . . . . . 92
3.7
Dendritic growth in directional solidication with a linear temperature gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1
Comparison of exact and numerical interface velocity for one-dimensional
boiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2
Comparison of exact and numerical uid velocities for one-dimensional
boiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3
The density prole illustrates the nite numerical thickness of the
interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4
Comparison of exact and numerical pressure for one-dimensional
boiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.5
A lm boiling simulation. . . . . . . . . . . . . . . . . . . . . . . . . 125
4.6
The pressure eld at t = 10. . . . . . . . . . . . . . . . . . . . . . . 126
4.7
Grid resolution study for lm boiling. . . . . . . . . . . . . . . . . . 127
4.8
A lm boiling simulation for l=v = 100. . . . . . . . . . . . . . . . 128
viii
4.9
Comparison of heat transfer results from two-dimensional numerical
simulations against a correlation of experimental data. . . . . . . . . 129
4.10
Rapid evaporation from an initially nearly planar interface in microgravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.11
A plot of interface length vs. time illustrates the energetic growth
of the interface instability. . . . . . . . . . . . . . . . . . . . . . . . 131
4.12
The growth of interface instabilities on a bubble growing in a superheated liquid in microgravity. . . . . . . . . . . . . . . . . . . . . . 132
4.13
The interface and temperature contours at t = 0:03. . . . . . . . . . 133
4.14
The interface and velocity eld at t = 0:03. . . . . . . . . . . . . . . 134
4.15
Bubble breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
ix
NOMENCLATURE
l identity tensor
A surface area
a
iteration constant in modied Newton iteration
C
C~
solute concentration
transformed solute concentration
C~f transformed solute concentration at the interface
Cl liquid solute concentration
Co initial solute concentration
Cs solid solute concentration
c
(volumetric) specic heat
cl
(volumetric) liquid specic heat
cs
(volumetric) solid specic heat
cv (volumetric) vapor specic heat
D chemical diusivity
Dl liquid chemical diusivity
D~ l transformed liquid chemical diusivity
Ds solid chemical diusivity
D~ s transformed solid chemical diusivity
d
interface point spacing
x
E(V) residual error vector in Newton iteration
F
Mullins-Sekerka stability function,
force per unit volume,
numerical distribution function
Fr
Froude number, Fr = Gl=Uo2
f
force per unit area
G
temperature gradient,
Earth gravity,
free energy
Gl
liquid temperature gradient
Gs
solid temperature gradient
G
grid-gradient eld
g
x,component function of the parametric interface representation
g
gravity vector
Hx
domain width
Hy
domain height
h
mesh size,
y,component function of the parametric interface representation
I (x; t) indicator function
i
J
unit vector in x,direction
Ja
Jakob number, Ja = lcl (T1 , Tv ) =v Lo
j
unit vector in y,direction
K
thermal conductivity
Jacobian in Newton iteration
xi
Kl
liquid thermal conductivity
Ks
solid thermal conductivity
Kv
vapor thermal conductivity
k
partition coecient k = Cs(xf (t))=Cl(xf (t))
L
(volumetric) latent heat
Lo
(volumetric) latent heat at equilibrium phase change temperature
Le
Lewis number, Le = clDl =Kl
l
length scale
M
initial interface symmetry mode,
mass transfer at interface per unit volume
Mo Morton number, Mo = 4l G= 3 l
Mov Morton number based on vapor properties, Mov = 4v G= 3v
m
liquidus slope,
mass transfer at interface per unit area
N
number of interface points
Nu
Nusselt number, Nu = Kl qw =Kv Tw
n
normal vector
nf
initial interface perturbation mode
nk
symmetry mode of #n(n)
ns
symmetry mode of n(n)
P
pressure
Pf
interface pressure
Pl
liquid pressure
Pv
vapor pressure
xii
P1
ambient pressure
Pe
solutal Peclet number, in chapter III, Pe = Vol=Dl ,
thermal Peclet number, in chapter IV, Pe = lclUol=Kl
Pr
Prandtl number, Pr = lcl=Kl
Prv
Prandtl number based on vapor properties, Prv = v cv =Kv
p
minimum interface point separation distance
Q
heat source per unit volume
QL
line heat sink strength
q
heat source per unit area
qw
wall heat ux
R
radius
Rb
initial interface radius perturbation
Rf (t) interface radius
Ro
average initial interface radius
R
critical nucleation radius
R
vector parametric interface representation
Ra
Rayleigh number, Ra = 83 [Mov (l=v , 1)],1=2 Prv
Re
Reynolds number, Re = lUol=l
r
radial coordinate
S
Solute source per unit volume,
Sekerka number, S = Pe (Ks =Kl + 1) = (GsKs =Kl + Gl )
St
Stefan number,
in chapter II, St = cs (T1 , Tm)=Lo ,
in chapter III, St = clTo=Lo
xiii
s
Solute source per unit area,
surface distance
T
temperature
Tf
interface temperature
Tl
liquid temperature
Tm
equilibrium freezing temperature
Ts
solid temperature
Tv
equilibrium vaporization temperature,
vapor temperature
Tw
wall temperature
T1
ambient temperature
To alloy freezing range, To = mCo(k , 1)=k
t
time
t
tangent vector
Uo
velocity scale
u
parameter for interface representation
u
uid velocity vector
V
normal interface velocity, V = (dxf =dt) n
V
interface velocity vector
Vmax maximum normal interface velocity
Vo
directional solidication translation velocity
v^
specic volume, v^ = 1=
v^l
liquid specic volume
v^v
vapor specic volume
xiv
We Weber number, We = l Uo2l=
w
wf
wl
wv
uid mass ux vector, w = u
X
cartesian coordinate
x
coordinate position vector
xc
x,coordinate of initial interface centroid
x
interface coordinate position vector
Y
cartesian coordinate
yc
y,coordinate of initial interface centroid
f
interface uid mass ux vector
liquid mass ux vector
vapor mass ux vector
Greek Letters
thermal diusivity, = K=c
l
liquid thermal diusivity
s
solid thermal diusivity
surface tension coecient
(n) anisotropic surface tension coecient
delta function,
interface perturbation amplitude
tolerance
f
amplitude of initial interface perturbation
k
amplitude of #n(n)
xv
s
amplitude of n(n)
constant in the exact solution of solidication by a line heat sink
angle
o
initial interface rotation angle
#
nondimensional inverse kinetic mobility,
in chapter II, # = Ks='Lo l,
in chapter III, # = Vo='To,
in chapter IV, # = lclUo =v Lo'
#n(n) nondimensional anisotropic inverse kinetic mobility
curvature
wavelength
dynamic viscosity
l
liquid dynamic viscosity
v
vapor dynamic viscosity
density
f
interface density
l
liquid density
v
vapor density
interface perturbation amplitude growth rate
nondimensional capillary parameter,
in chapter II, = cs Tm=L2o l,
in chapter III, = kTmVo =LoToDl,
in chapter IV, = clTv =v L2o l
xvi
n(n) nondimensional anisotropic capillary parameter
phase eld variable
k
symmetry angle of #n(n)
s
symmetry angle of n(n)
'
kinetic mobility
'(n) anisotropic kinetic mobility
!
wave number
xvii
ABSTRACT
A computational methodology based on a front tracking/nite dierence method
is developed for the direct simulation of two-dimensional, time-dependent phase
change processes involving: (1) pure material solidication, (2) alloy solidication
and (3) liquid-vapor phase change with uid ow. The method is general in the sense
that large interface deformations, topology change, latent heat, interfacial anisotropy
and discontinuities in material properties between the phases are directly incorporated into the problem formulation and solution technique.
For simulations of solidication of pure materials the accuracy of the front tracking method is veried through comparison with exact solutions. Convergence under
grid renement is demonstrated for unstable solidication problems. Experimentally observed complex dendritic structures such as liquid trapping, tip-splitting,
side branching and coarsening are reproduced. It is also shown that a small increase
in the liquid to solid volumetric heat capacity ratio markedly increases the solid
growth rate and interface instability.
For the directional solidication of binary alloys, the coupled solute and energy
equations are considered. Convergence under grid renement is demonstrated and
results for the growth of instabilities is shown to be in close agreement with a linear
stability analysis. During the transient development of cellular and dendritic structures realistic phenomena such as liquid trapping, coarsening and droplet detachment
from deep cellular grooves are observed. These simulations also predict a variety of
xviii
microstructural solute segregation patterns such as necking, coring and banding.
For liquid-vapor phase change with uid ow the coupled Navier-Stokes and energy equations with interphase mass transfer are solved. The method is validated
through comparison with an exact one-dimensional solution and by grid resolution
studies. In lm boiling a vapor layer adjacent to a heated surface undergoes a
Rayleigh-Taylor instability with vaporization at the interface. Pinch o of a vapor
bubble causes hot vapor from regions near the wall to be convected upward in the
rising bubble. Heat transfer results are compared with a correlation of experimental data. Simulations of the rapid evaporation of a highly superheated liquid under
microgravity conditions demonstrate the energetic growth of instabilities from planar and circular interfaces. The formation of highly convoluted interfaces leads to
enhanced evaporation and explosive growth.
xix
CHAPTER I
INTRODUCTION
1.1 Motivation
From boiling water on a stove, to a reading lamp supplied with electricity generated by condensing steam, to the melting of ice cubes in a glass of lemonade, phase
change is a phenomenon that is so frequently encountered in daily life that its occurrence and its contributions to the products and services that one uses is taken for
granted. Phase change plays an important role in the power, chemical, petroleum
and electronics industries among others.
Nearly all materials of engineering interest have at some point solidied from a
liquid state. In materials processing it is the exact nature of the solidication process
that determines the microstructure and thus the physical properties and usefulness
of the solid material. Solidication is usually inuenced by a combination of phase
change, uid ow and heat transfer. The electrical properties of semiconductors and
the mechanical properties of single crystal turbine blades are closely coupled to uid
convection and processing conditions during directional solidication. Research on
advanced materials and manufacturing processes such as rapid solidication or spray
casting for the production of net or near-net shaped components depends on the
ability to control and understand uid ow during solidication.
1
2
The power generation industry takes advantage of the high heat transfer rates
associated with phase change in boiling to extract energy from solar, fossil and
nuclear fuels. Designers of energy generation systems for spacecraft must deal with
the added complication of handling low boiling point cryogenic uids in the absence
of gravity. Understanding the behavior of liquid-vapor interfaces is a fundamental
requirement for safe spacecraft operation. Accidental vapor explosions due to sudden
depressurization or overheating of uid lled tanks are a serious safety concern both
on Earth [1] and in space (oxygen tank explosion aboard Apollo 13). Experimental
investigations of boiling in microgravity [2, 3] have been prompted by this concern
and the need to more fully understand vapor bubble behavior in the absence of
gravity.
Despite the ubiquitous nature of phase change the details of the phase change
process are far from being completely understood. Experimental investigations of
phase change in solidication and boiling are generally dicult due to the large range
of important time and length scales involved. In addition, the harsh thermal and
chemical environments in directional solidication furnaces make it nearly impossible
to obtain direct measurements of the solidication process. Numerical simulations
hold the promise to complement experimental investigations and provide information
that is hard to measure. Although computations of industrial scale processes are
currently out of reach, a fundamental understanding of the complex physics at the
small scale can provide much needed insight to larger problems. By understanding
this small scale information it is hoped that progress can be made toward the longterm goal of providing quantitative predictions for linking operating conditions to
large scale aspects of heat exchanger design and eciency or to mechanical properties
of solidied materials, for example.
3
1.2 The State of the Art
Investigations of problems involving complex moving interfaces have remained
a dicult challenge. Theoretical and experimental studies have laid the necessary
groundwork but it is clear that computational modeling oers the promise of helping
to provide accurate predictions of physical phenomena where complex interactions
among many eects such as uid ow, surface tension, heat transport and phase
change cannot be ignored.
Numerical techniques for solidication problems without uid ow have been
available for decades. Enthalpy methods [4{10] appear to be the most advanced
and most popular for large scale problems where the details of the solidication
microstructure are not of interest. Simulations of small scale microstructural features are much more dicult due to the very large range of time and length scales
involved. Complex two-dimensional simulations for both pure materials and alloys
are, however, starting to become commonplace. Interface capturing methods such
as the phase-eld method [11{31] and methods using boundary tted grids [32{45]
are receiving a great deal of attention and have been implemented in two- as well as
some three-dimensional simulations.
Numerical techniques for incompressible, multi-uid ows without phase change
have advanced to the point where relatively complex two-dimensional simulations are
commonplace. Common approaches use interface capturing [46{54], boundary tted
grids [55{57], Lagrangian methods [58{61], boundary integral methods [62, 63] and
front tracking [64{68]. A number of three-dimensional simulations have also been
performed using front tracking [64{66,69{74].
One of the next unresolved challenges in numerical simulations is the combination
4
of uid ow with phase change [75{77]. The numerical solution of the phase change
problem with uid ow is among the more complex transport processes encountered
in engineering applications since it is necessary to model the interactions of uid
ow with heat and mass transfer across moving phase interfaces. Computations of
this problem are still far behind what is possible for multi-uid ows without phase
change. Typical numerical models use an assumed interface shape [78{86] or do not
consider ows in which the interface deforms greatly [87{89]. Various simplications
concerning surface tension, uid viscosity, vapor phase velocity and temperature are
also usually made.
The present state of the art in direct numerical simulations of phase change can
be summarized as follows:
Pure Material Solidication.
Phase-eld methods have been developed and implemented in unsteady, twodimensional as well as some three-dimensional computations of complex dendritic structures. The interface is captured using only a regular, stationary grid
and the method is thus relatively easy to implement. Boundary tted moving
nite elements have also begun to be used in three-dimensional computations of
dendrites. This method may possibly oer the highest accuracy at the expense
of tremendous computational complexity.
Alloy Solidication.
Boundary tted nite element and nite dierence grids have been used in
steady-state, two-dimensional and limited three-dimensional simulations of cellular growth in nonisothermal, directional solidication. Unsteady simulations
using adaptive moving nite elements have been limited to small interface
5
distortions. Phase-eld methods have been extended to model the unsteady
solidication of alloys but have so far only been implemented for isothermal
systems.
Liquid-Vapor Phase Change with Fluid Flow.
For ows with only small interface distortions, the mass, momentum and energy equations with interphase mass transfer have been solved using a twodimensional, moving mesh, nite volume method [87] and also for multiple
components using a boundary-tted nite element method [88].
Overall, progress is being made in simulating the qualitative features of solidication microstructures. However the capability for quantitative predictions of realistic
situations has remained elusive. For liquid-vapor phase change with uid ow, direct
computations have been limited to simple problems and interface geometries.
1.3 Contributions Made in this Work
A computational methodology has been developed for the direct simulation of
time-dependent phase change processes involving: (1) pure material solidication,
(2) alloy solidication and (3) liquid-vapor phase change with uid ow. The method
is general in the sense that large interface deformations, topology change, latent heat,
interfacial anisotropy and discontinuities in material properties between the phases
are directly incorporated into the problem formulation and solution technique.
The methodology is an extension of a front tracking/nite dierence technique
developed for multi-uid ows without phase change by [65, 66]. Those front tracking concepts have been reimplemented and modied in order to incorporate phase
change. The generality and modularity of the front tracking method is emphasized
by the fact that the concepts and computational algorithms for front tracking have
6
been written and used without modication for all three of the phase change problems considered here. These concepts are collected and described in the appendix.
Our contributions to the state of the art in direct numerical simulations of phase
change are:
Chapter II: Pure Material Solidication.
An alternative method has been developed for time-dependent, two-dimensional
dendritic solidication of pure materials. The ability of the method to handle
the general case of unequal material properties allowed the exploration and
identication of a previously unreported increase in instability associated with
unequal specic heats between the liquid and solid phases.
Chapter III: Alloy Solidication.
A method for time-dependent, two-dimensional solidication of dilute binary
alloys has been developed. This appears to be the rst method which solves
the coupled, time-dependent solute and energy equations including the eects
of large interface deformations, topology change, latent heat, anisotropic capillarity and interface kinetics and discontinuities in material properties between
the liquid and solid phases. The major contribution is that latent heat is not
neglected and a known temperature or an isothermal system is not assumed.
The method is used here for simulations of directional solidication.
Chapter IV: Liquid-Vapor Phase Change with Fluid Flow.
A method for time-dependent, two-dimensional liquid-vapor phase change with
uid ow has been developed. This appears to be the only method available
at present which takes into account mass, momentum and energy transport
with interphase mass transfer while allowing large interface deformations and
7
topology change. It also accounts for latent heat, surface tension and unequal
material properties between the liquid and vapor phase. The method is used
here for simulations of lm boiling in Earth gravity and rapid evaporation in
microgravity.
CHAPTER II
PURE MATERIAL SOLIDIFICATION
2.1 Introduction
Dendritic growth of crystals into an undercooled liquid is a very common form
of solidication in castings, ingots and welds. The microstructure produced upon
solidication determines the qualities of the solidied raw material and often the
nished product. This problem has attracted much interest for several years and has
been motivated by the desire to predict crystalline microstructure in designing solidication methodologies for advanced materials in the aerospace and semiconductor
industries, for example. Protein crystallization and igneous rock formation are just
two examples of problems where researchers in elds as diverse as medicine and geology also stand to benet from a better understanding and control of crystal growth.
Despite the large volume of literature dealing with dendritic growth, the problem is
still not well understood, even for the simplest case of solidication of a pure substance. Mathematical theories and numerical investigations have had only limited
success in comparison with experiments. Here, a numerical method for solidication
problems based on a simple nite dierence approximation of the heat equation and
explicit tracking of the liquid-solid interface is presented. The method is general
in the sense that it can handle discontinuities in material properties between liquid
8
9
and solid phases, interfacial anisotropy and topology changes. An overview of previous research on theories and simulations of morphological instabilities and dendritic
solidication is given rst, in order to put the present work into perspective.
The process of solidication of a pure substance can occur in either a stable or an
unstable manner. Stable solidication, classically called the Stefan problem, is characterized by conduction of heat away from the solid-liquid interface through the solid.
The interface generally remains smooth; any protrusions of the solid into the liquid
are retarded. Stable solidication is dominated by heat diusion while surface tension
and interface kinetic eects are negligible. Analytic solutions of the Stefan problem
for simple geometries are well-known (see, for example, [90]). Numerical methods
for the solution of more complex situations include the popular enthalpy method [4]
and methods using a coordinate transformation to immobilize the boundary. In the
enthalpy method the interface is not explicitly tracked but must be determined after
the solution has been obtained. In this respect the method is easy to use if precise
knowledge of the interface location is not critical. It has received widespread use
in industrial applications where the phase change occurs over a temperature range
and the melt/solid interface can be described as a mushy zone. However, Voller et
al. [5] demonstrate that the enthalpy method produces non-physical features when
the melting temperature is sharply dened. Voller and Cross [6] propose an extension to the conventional enthalpy method which eliminates this problem and they
demonstrate its applicability to one- and two-dimensional problems. More recently,
Swaminathan and Voller [7] have developed a general enthalpy method which encompasses both the source based and the apparent specic heat enthalpy methods.
From the general method they identify an optimal enthalpy scheme for a range of
two-dimensional phase change problems. Comini et al. [8] compare the performance
10
of several enthalpy-based algorithms. Voller and Swaminathan [9, 10] review xed
grid techniques for phase change problems and enthalpy methods in particular. Other
methods for solving stable solidication problems include inverse methods such as the
isotherm migration method [91], an inverse nite element method by Alexandrou [92]
and nite element methods using a deforming mesh [93].
Unstable solidication of a pure substance takes place when the liquid is cooled
below its equilibrium solidication temperature. Heat is conducted away from the
solid-liquid interface through the liquid. Any local protrusion on the interface that
extends into the liquid will be enhanced since the magnitude of the temperature
gradient at the protrusion is greater than that at adjacent portions of the interface.
The process is inherently unstable and the protrusion will grow until constrained
by surface tension and interface kinetic eects. Morphologically complex dendritic
structures result from this competition between surface tension and undercooling.
Understanding and modeling the mechanisms which produce these structures has
been the focus of much research.
The primary instability mechanisms of a steadily advancing planar interface were
analyzed by Mullins and Sekerka [94] and Voronkov [95]. A similar linear stability
analysis was performed for growing spheres by Mullins and Sekerka [96] using a
quasi-stationary assumption that has been extended to include interface kinetics
[97,98]. Steady-state models of dendrite growth are based on Ivantsov's solution [99]
of the heat transport equation for a paraboloidal, isothermal interface growing at
constant velocity into a uniformly undercooled liquid. For a given undercooling, an
innite number of solutions are given by combinations of the growth speed, V , and
the tip radius, R according to the relationship V R = const. However, this set of
solutions is clearly inconsistent with the thermodynamic constraint of a minimum
11
radius below which no growth can occur. The Ivantsov model also does not take into
account eects due to surface tension and interface kinetics. Temkin [100], Bolling
and Tiller [101], Trivedi [102], and Glicksman and Schaefer [103] addressed these
problems by including surface tension eects. Similar to Ivantsov's result, an innite
number of solutions are given in terms of V , R combinations. However, for a given
undercooling the curves exhibit a maximum velocity which was thought to be the
unique operating state of the dendrite. Unfortunately, the tip radii predicted by the
maximum velocity hypothesis do not agree well with experimental observations [104].
Nash and Glicksman [105] formulated a self-consistent free boundary problem in the
form of a non-linear integro-dierential equation. Solution of this equation [106]
yielded only a slight modication to Ivantsov's paraboloid shape and results similar
to the maximum velocity hypothesis.
Time-dependent theories for morphological instability were developed in response
to the shortcomings of the steady-state models. The concept of marginal stability
was established by Langer and Muller-Krumbhaar [107{109]. They assumed that
the operating state of the dendrite tip lies at the margin of the linearly stable and
unstable states. The analysis resulted in a second relation between V and R which
together with Ivantsov's solution determines a unique operating state. The marginal
stability theory has come under question in light of recent measurements of dendrite
tip velocities in pure succinonitrile in microgravity by Glicksman et al. [110, 111].
In addition, Glicksman and Marsh [104] contend that the marginal stability theory
cannot be considered as providing a fundamental theory of dendritic crystallization
due to limitations concerning the validity of both its theoretical and mathematical
foundations.
More recent theories which attempt to address the concern over marginal stabil-
12
ity resulted in advanced analytical and numerical approaches to the solution of the
Nash-Glicksman integro-dierential equation and variants of this equation. The role
of anisotropy of the interfacial energy in providing a selection mechanism for the dendrite operating state was used to nd solutions of the anisotropic form of the NashGlicksman equation. These so-called microscopic solvability theories are reviewed
by Kessler et al. [112, 113], Langer [114], Pelce [115]. and Kurz and Trivedi [116].
These theories and a related method by Miyata et al. [117] are not supported by
recent experiments on camphene, pivalic acid and succinonitrile by Rubinstein and
Glicksman [118,119] and there is still a debate over whether anisotropy of the interfacial energy provides a fundamental physical basis for the dendrite operating state
selection [104].
Numerical methods for dendritic solidication problems are not as common as
those for Stefan problems due primarily to the diculties involved in handling the
complex, unstable interface shapes. Several numerical simulations, mostly in twodimensions, have however, been successful in obtaining qualitative agreement with
observed dendritic structures. Sullivan et al. [120, 121] use a nite element method
with a deforming mesh and a careful numerical treatment of interface curvature
to perform two-dimensional simulations. They have also modied this method to
include anisotropic material properties [122]. In spite of the two-dimensional limitations, they obtained good agreement with experimental observations of dendrite
tip velocities as a function of tip curvature. Tacke [123] applied a nite dierence
enthalpy method to the 2-D dendritic solidication problem. Although his results
exhibit qualitatively realistic phenomena, the four-fold symmetry of the dendritic
structures in his simulations is due to a canalization eect of the grid and not to
any physical anisotropy. Sethian and Strain [124] use a level set approach to de-
13
termine the solid/liquid boundary and include eects of anisotropic surface tension
and interface kinetics. Almgren [125] uses a variational algorithm to perform similar
computations. The interface is tracked and its shape is determined by minimizing
an energy functional made up of bulk and surface energy contributions. The interface temperature condition is only approximately satised at each time step in this
method. Roosen and Taylor [126] introduce a front tracking scheme which avoids
direct computation of interface curvature by assuming that the interface is a polygon. They admit, however, that their method contains an inherent grid induced
anisotropy which is present even at high grid resolutions. Shyy et al. [127] have used
an interface-tracking method for the solution of stable and unstable solidication
problems in which they map the governing equations into curvilinear coordinates.
For the unstable problem, they use scaling arguments to decompose the domain into
an inner region where interface perturbations develop and an outer region where
the interface is assumed to be planar. The two regions are coupled by matching
of the boundary conditions. Brattkus and Meiron [128] have developed an ecient
algorithm to compute one-dimensional free-boundary problems using a boundary integral formulation. Their method can handle the general problem of unequal thermal
diusivities between the liquid and solid phases. They have applied the method to
study oscillatory instabilities in rapid directional solidication.
Phase-eld models and their numerical implementation are currently the subject
of considerable interest. Langer [11], Fix [12], Caginalp [13,14], Caginalp and Fife [15]
and Collins and Levine [16] have extensively studied and adapted this method. A
phase-eld variable is postulated which identies the phase of a point in the domain. If the point lies in the liquid region, = 0; if the point lies in the solid
region, = 1. Values of between zero and one represent points that lie in the
14
interface. An evolution equation for this scalar function consistent with the second
law of thermodynamics is coupled to a modied heat equation. Solution of this
system of equations provides values of temperature and the phase-eld variable and
thus implicitly the interface location. The main advantage of this approach is that
complex topology changes are easily handled since there is no need to explicitly track
the interface or even provide interfacial boundary conditions. The disadvantage of
this method is in relating the parameters in the evolution equation for to phenomenological parameters such as surface tension and interface kinetic coecient.
Caginalp [14] has shown that the classical phenomenological boundary conditions of
solidication are recovered under certain limits of the phase eld equations. Caginalp
and Socolovsky [17] conducted spherically symmetric calculations using the phaseeld model. Their work provided numerical verication of the concept of a critical
radius and qualitative agreement with single needle crystals. Impressive two- and
three-dimensional numerical computations by Kobayashi [18{20] reveal qualitatively
correct large scale features of dendritic structures. However, the simulation parameters had to be carefully adjusted to produce the desired structures and Wheeler [22]
demonstrates that small-scale features of Kobayashi's calculations such as liquid
trapping and tip splitting events are crucially dependent on the mesh used. There is
still debate over the thermodynamic basis of some of the phase-eld models currently
in use and their relation to interfacial dynamics [21], [22]. Apart from phase-eld
methods they only other computations of three-dimensional dendrites is in a recent
work by Schmidt [32].
Penrose and Fife [21] and more recently Wang et al. [23] have developed thermodynamically consistent phase eld models based on an entropy functional. Wheeler et
al. [24] and Murray et al. [25] have used the phase eld model of Wang et al. to crit-
15
ically assess the computational viability of phase eld models. For two-dimensional
computations of anisotropic nickel dendrites they nd that their results are in good
agreement with the Ivantsov and microscopic solvability theories for a given phase
eld parameter which determines the interface thickness. The results, however, are
dependent on the interface thickness. They suggest that, at present, realistic phase
eld computations suer from the inability to suciently resolve the interface. An
investigation of the interface resolution problem in phase eld models using linear
stability analyses is the subject of recent work by Braun et al. [129].
Although signicant advancements in describing and understanding the mechanisms of morphological instability have been made in the past several decades, it
is clear that no single unifying theory is available which can accurately predict the
microstructure of unstable solidication. Numerical schemes are mostly limited to
simulations of the qualitative features of dendritic growth. The aim of the present
work is to provide a new numerical tool with which to study and identify the mechanisms of dendritic growth and instability under a large range of conditions. The
mathematical formulation of the governing equations and boundary conditions is
given in section 2.2. The numerical method described in section 2.3 is general in the
sense that it can easily handle discontinuous material properties between the liquid
and solid phases, topology changes and anisotropy of interfacial energy and kinetics.
A direct approach to the numerical simulation of the governing phenomenological
equations and interface conditions is taken. Solutions for the heat ow and interface
motion are coupled at each time step. front tracking explicitly provides the location of the interface at all times and the Gibbs-Thomson condition on the interface
temperature, Eq. (2.5), is also explicitly satised. In this way the introduction of
non-physical simulation parameters is avoided. Undercooling, surface tension, kinetic
16
mobility and the thermal conductivity and volumetric specic heat ratios between
liquid and solid directly control the solution. In section 2.4, results using this method
for both stable and unstable solidication problems are discussed. First, numerical
results are compared to an exact solution of the stable Stefan problem of solidication by a line heat sink. Then, for unstable dendritic solidication the validity of the
results is assessed through grid renement studies and comparison with theories for
nucleation and limiting solid fraction. The eect of discontinuous material properties
on the interface growth rate and stability is also identied.
2.2 Mathematical Formulation
Consider a square, wall bounded two-dimensional domain in which the solidication of a pure substance and the evolution of the liquid-solid interface is to be
described. In the dendritic solidication problem, a small seed of solid is introduced
into an undercooled liquid. Initially, the temperature everywhere in the solid is assumed to be equal to the equilibrium freezing temperature, Tm, and the temperature
in the surrounding liquid to be T1. Thus the liquid is undercooled by an amount
T1 , Tm. The densities of the liquid and solid phases are assumed to be equal
and constant. Volume contraction and expansion as well as uid convection eects
are thereby neglected. These eects are addressed in chapter IV for the problem of
liquid-vapor phase change with uid ow. The thermal conductivity and volumetric
specic heat of each phase are constant but not necessarily equal. A formulation employing a single heat equation can be written for both phases as long as the jumps
in material properties and the liberation/absorption of latent heat at the interface is
correctly accounted for. In conservative form the heat equation is
@ (cT ) = r K rT + Q
@t
(2.1)
17
where T is the temperature eld, and c and K are the volumetric specic heat
and thermal conductivity, respectively. Q is a volumetric energy source term which
accounts for the liberation or absorption of latent heat at the liquid-solid interface
Q=
Z
A
q (x , xf ) dA :
(2.2)
q is the energy source at the interface per unit area and (x , xf ) is a threedimensional delta function that is non-zero only at the interface where x = xf .
Since the above integral is over a surface, the source term, Q, is still a delta function.
The above formulation is not new and has been used, for example, by Lightfoot [130]
in analytic solutions to phase change problems and more recently in source based
enthalpy methods [7]. The treatment of the interface source term, Q, is of crucial
importance to numerical solutions which use the phase change formulation in Eq.
(2.1). The front tracking method presented here allows a detailed description of
the interface microstructure which is of primary interest in dendritic solidication
problems.
The expression for the interface energy source, q, is
q = LV :
(2.3)
V = (dxf =dt) n is the normal velocity of the interface where n is the normal to the
interface. L includes the eects of unequal specic heats,
L = Lo + (cs , cl) Tm
(2.4)
where Lo is the customary volumetric latent heat of fusion measured at the equilibrium melting temperature, Tm. cl and cs are, respectively, the liquid and solid
volumetric specic heats.
18
In addition, a temperature condition, the so-called Gibbs-Thomson condition,
must be satised at the interface. A thermodynamic analysis of phase coexistence at
a curved interface reveals terms in the Gibbs-Thomson condition which account for
the eect of discontinuous specic heat across the interface. This expression, derived
in [131], is
Tf , Tm + Tm (cLl , cs ) Tf ln TTf + Tm , Tf + (nL)Tm + 'V(n) = 0
o
m
o
(2.5)
where (n) is the anisotropic surface tension coecient given as a function of the
local surface normal orientation and is twice the mean interface curvature which is
positive when the center of curvature lies in the solid phase. The last term accounts
for the eect of anisotropic kinetic mobility, '(n), and is not included in [131].
It is intended to model the inherent non-equilibrium nature of the phase change
process. The assumption is made that kinetic eects are linearly proportional to
the interface temperature. At large undercooling and thus high growth velocity this
parameter adds a stabilization eect to the interface by depressing the local freezing
temperature. A small enough value of ' acts simply to suppress the growth of any
unstable protrusions of the interface.
Note that integration of Eq. (2.1) across the interface directly yields the correct
form of the interfacial energy balance (Stefan condition) [131]
[Ks rTs , Kl rTl] n = [Lo + (cl , cs) (Tf , Tm)] V
(2.6)
where Tf is the interface temperature, T (xf (t)), and Kl and Ks are the liquid and
solid thermal conductivities respectively. The term in brackets on the right side of
Eq. (2.6) can be thought of as a temperature dependent latent heat due to the eects
of discontinuous specic heat and the fact that the phase change generally occurs at
a temperature, Tf , dierent than the equilibrium melting temperature, Tm. Thus the
19
formulation in Eqs. (2.1)-(2.4) naturally incorporates the correct interfacial energy
balance without the numerical diculty of calculating temperature gradient values
on the interface.
If the specic heat is equal in both phases and in the absence of molecular kinetic
eects and anisotropy, Eq. (2.5) reduces to the classic Gibbs-Thomson condition
m
Tf = Tm , T
L :
o
(2.7)
In order to dierentiate liquid and solid material regions an indicator function,
I (x; t), is used which is similar to the phase-eld variable in phase-eld models that
has the value 1 in the solid phase and 0 in the liquid phase. Unlike the phase-eld
variable, I (x; t) is constructed from the known position of the interface rather than
used to determine the position of the interface. The numerical construction of the
indicator function is discussed in appendix A.4. This function enables the evaluation
of the values of the material properties at every location by
c(x) = cl + (cs , cl) I (x; t) ;
(2.8)
K (x) = Kl + (Ks , Kl) I (x; t) :
(2.9)
The governing equations and boundary conditions can be made dimensionless by
scaling length by a suitable length scale, l, time by Ks =cs l2, velocity by cs l=Ks, Q
and q by cs l=Ks Lo and temperature (measured from Tm) by Lo=cs . The volumetric
specic heat and thermal conductivity elds are scaled by cs and Ks respectively
c = ccl + 1 , ccl I (x; t) ;
s
s
Kl + 1 , Kl I (x; t) :
K=K
Ks
s
Eqs. (2.1)-(2.5) become
@ (cT ) = r K rT + Q
@t
(2.10)
(2.11)
(2.12)
20
Z
q (x , xf ) dA
q = LV
Lo
Tf + ccl , 1 Tf2 + n(n) + #n(n)V = 0 :
Q=
A
s
(2.13)
(2.14)
(2.15)
The above set of nondimensional equations, Eqs. (2.10)-(2.15), is the mathematical
formulation of the solidication problem which is solved numerically.
In deriving Eq. (2.15) the rst order approximation ln(1 + z) z for small z
is used in the specic heat term. In this case, z = (Tf , Tm) =Tm. A conservative
estimate for the maximum value of z is at nucleation where the phase change temperature is close to the bulk liquid temperature, Tf T1. Then, maximum observed
values for z are 0.14 for water, 0.04 for succinonitrile and 0.13-0.25 for most pure
metals [132] which leads to an error in the logarithm approximation of roughly 10%
in the worst case and only a few percent for more typical undercoolings. Far from
nucleation conditions, where the simulations here are run, Tf is closer to Tm than it
is to T1 and the logarithm approximation becomes even better.
In Eq. (2.15) the term involving the specic heat ratio is second order in the
local interface temperature, Tf , and may usually be neglected except at large undercooling. However, note that for hypercooled situations this term is signicant. In
the simulations this term is included since it may have some eect in the parameter
range of the simulations performed here.
The functional form of the two-dimensional anisotropic capillary parameter, n (n),
and inverse kinetic mobility, #n(n), is similar to that used by Almgren [125]
n() = 1 + s 83 sin4 12 ns ( , s) , 1
#n() = # 1 + k 83 sin4 21 nk ( , k ) , 1
(2.16)
(2.17)
21
where
= csLT2ml o
(2.18)
is the isotropic capillary parameter and
Ks
# = 'L
l
o
(2.19)
is the isotropic inverse kinetic mobility. The constants s, k determine the magnitude
of anisotropy, ns and nk the mode of symmetry of the crystal and s and k determine
the angle of the symmetry axis with respect to the x,axis. The idea behind this
choice of function is to model a crystalline material with a sharp cornered polygonal
shape. For example, for s = 1, n () ranges from 0 to 8=3 with a fourth order
minimum at ( , s ) = 2n=ns . For ns = 4 the resulting shape would be four-fold
symmetric.
2.3 Numerical Method
The numerical technique used for the simulations is based on the front tracking
method developed for multi-uid ows by Unverdi [64] and by Unverdi and Tryggvason [65, 66] in three-dimensional simulations of rising and colliding bubbles. It has
been used to study the rise of contaminated bubbles (Jan and Tryggvason [133]),
the axisymmetric collision of two drops (Nobari et al. [134]), three-dimensional drop
collisions (Nobari and Tryggvason [74], the thermal migration of a two-dimensional
bubble cloud (Nas and Tryggvason [135]) as well as other problems. Here, the procedure will be described as it is applied to moving boundary problems in solidication
and specically to the numerical solution of Eqs. (2.12)-(2.15).
22
2.3.1 Discretization
In the numerical solution, a square, stationary, regular grid of mesh size h is used
and the heat equation, Eq. (2.12), is discretized using a conservative, second order,
centered dierence scheme for the spatial variables and an explicit, rst order, forward Euler time integration method. The method is traditionally called the forward
in time, centered in space, or FTCS, scheme. (Note that the interface-tracking is
independent of the method used to solve the governing equations on the stationary
grid and thus the method is not restricted to the use of nite dierences but may
be implemented using nite elements or nite volumes. Second order time integration can also be easily implemented for increased accuracy.) A variable time step
that depends on the mesh size, h, and the normal interface velocity, V , is used. Its
value is determined such that two criteria are met. First, the maximum value of
the time step must satisfy the stability requirement for the two-dimensional FTCS
scheme. For stability, the time step, t, is required to satisfy t h2=4 where is the thermal diusivity of the solid or liquid, whichever is larger. Second, in order
to ensure that the interface does not move a distance larger than about 1/10 of a
mesh block in one time step, the time step is also restricted to t h= (10 jVmaxj),
where jVmaxj is the magnitude of the maximum value of the interface velocity. (1/10
was found to provide sucient stability.) Generally, the rst criterion determines
the time step throughout most of the computation except at early times when the
interface motion is rapid.
The interface is explicitly tracked by using separate, non-stationary computational points connected to form a one-dimensional front which lies within the twodimensional stationary mesh. The details of the interface representation, the distribution and interpolation technique used to transfer information between the front
23
and the stationary grid and the construction of the indicator function, I (x; t) are
common to the three problems studied in this thesis and are discussed separately in
the appendix.
2.3.2 Solution Procedure
In order to begin the computation an initial interface shape is specied. From
this shape the indicator function is constructed as described in appendix A.4 and
the specic heat and thermal conductivity elds are determined from Eqs. (2.10)
and (2.11). In a similar manner the indicator function is used to specify the initial
temperature eld,
T (x; t = 0) = St (1 + I (x; t)) ;
(2.20)
where the Stefan number, St = cs(T1 , Tm)=Lo , is the nondimensional undercooling.
This initial temperature eld is of course only an approximation intended to model
the sudden introduction of a small seed of solid into an undercooled liquid. (The
Gibbs-Thomson condition, Eq. (2.15), is not necessarily satised for the initially
specied interface shape and temperature eld but it is satised after the rst time
step has been taken. Since the computation utilizes a variable time step, the step is
initially set to be quite small. The result after this rst small time step can then be
eectively considered the proper initial condition which satises the Gibbs-Thomson
condition.)
Given the initial interface shape, temperature and material property elds, the
solution algorithm proceeds iteratively through the following steps:
1. With the estimate for the updated normal velocity, V (t +t), from Eq. (2.21),
the heat source at the interface, q, is calculated from Eq. (2.14) and is distributed to the stationary grid using Eq. (A.11).
24
2. Using the updated velocity, the interface is advected to a new position by
V = (dxf =dt) n.
3. The indicator function is constructed at this new interface position by solving
Eq. (A.17) and the specic heat eld is found using Eq. (2.10) .
4. With appropriate wall boundary conditions, Eq. (2.12) is solved for the temperature eld at time t + t by a conservative FTCS scheme.
5. This temperature eld is interpolated by Eq. (A.12) onto the interface at its
new position to nd the temperature at each point on the interface.
6. If the Gibbs-Thomson condition, Eq. (2.15), is satised then the thermal
conductivity eld is updated to the new interface position by Eq. (2.11) and
the computation proceeds to the next time step. Otherwise, a new estimate for
the updated normal velocity, V (t + t), is found at each interface point using
Eq. (2.21) and the procedure returns to step 1.
In the last step, the new estimate for V is found by an iterative method. In
one-dimensional problems, where the interface is only one point, the bisection or
secant methods work successfully. In two dimensions, a multi-dimensional iterative
method must be used since the interface now consists of many points. Here a simple
iteration scheme for nonlinear sets of equations is described which appears to work
well. In general, if the interface temperature found in step 5 is substituted into Eq.
(2.15) the right hand side of this equation will not equal zero but some residual error,
E(V). In order to make this error go to zero and thus satisfy Eq. (2.15) a variation
of the Newton iteration method is used.
25
2.3.3 Modied Newton Iteration
In matrix form, the Newton iteration updates the unknown velocities at each
point by the equation,
Vl+1 = Vl , [J],1 El Vl ;
(2.21)
where l is the iteration index. V and E are, respectively, the N 1 column vectors of
interface velocities and errors at each point. N is the number of interface points. The
Jacobian, J, is the N N matrix of partial derivatives of the error with respect to
the velocities, Jmn = @Em=@Vn . Since these derivatives are dicult to calculate and
the subsequent matrix inversion is computationally expensive, a dierent Jacobian
is used which has the simple form,
J = a,1l ;
(2.22)
where l is the identity matrix and a is a constant. This constant determines the rate
of convergence of the iteration. In the code a is adjusted manually during the rst few
time steps until an optimum rate of convergence is achieved. At the optimum value of
a, which is dierent for dierent physical parameters, the iteration converges rather
quickly to a tolerance of = 10,5 in 3 to 10 iterations. The tolerance is calculated
by
N
1
(2.23)
= N Vkl+1 , Vkl :
k=1
Optimum values for a range roughly between 400 and 800 depending on the physical
X
parameters.
26
2.4 Results and Discussion
2.4.1 Stable Solidication
The two-dimensional Stefan problem of solidication in the plane due to a continuous line heat sink was solved numerically and compared with the exact analytical
solution given by Carslaw and Jaeger [90] in one-dimensional axisymmetric coordinates
2
T (r; t) = Q4L Ei ,4rt , Ei , 2
"
!
#
(2.24)
in the solid region and
Ei
(
,
s r2 =4l t)
T (r; t) = 1 , Ei (, 2= )
s
l
(2.25)
in the liquid region. QL is the strength of the line heat sink, is the thermal
diusivity and Ei is the exponential integral. The heat source at the circular interface
and its radius are
q(t) = 1
St t 2
where the constant is the root of
QL
= 4e2
Rf (t) = 2t 21
"
,Kl e, = + 2
Ks Ei (,s 2=l) St
s
2
l
#
(2.26)
(2.27)
As stated earlier, many researchers have performed numerical simulations of stable solidication using a variety of methods. The purpose of these computations
is to demonstrate the accuracy and robustness of the front tracking method. The
problem of solidication due to a line heat sink poses an especially rigorous test of
the numerical method due to the necessity of accurately predicting the location of
the moving phase change interface and temperature eld in the presence of large
temperature gradient variations from innity at r = 0 to nearly 0 at the boundary
of the computational domain. In these gures comparisons of numerical and exact
27
solutions for a variety of physical conditions and numerical parameters are presented.
The initial and boundary conditions on the temperature for the numerical solution
correspond to the exact solution at each time step. In order to avoid computation of
the innitely negative temperature at r = 0, this point was placed in the center of a
mesh block. The four points at the vertices of this block were maintained at the exact
solution temperatures. For the stable solidication problem, Eqs. (2.10)-(2.15) are
solved as described in the previous section. To be consistent with the exact solution
eects due to surface tension, interface kinetics and liquid/solid specic heat ratio
are ignored in Eqs. (2.14) and (2.15). (The eect of the specic heat ratio is still
included in Eq. (2.12) in the non-dimensional specic heat eld, c.) Eqs. (2.14) and
(2.15) for the stable problem become
q=V ;
Tf = 0
(2.28)
In Fig. (2.1a) the average radius of all the interface points is plotted at each time for
three dierent grid resolutions: 1010, 2020 and 5050. For a heat sink strength,
QL = 10, and Stefan number, St = 1, the solution converges to the exact solution
for increasing grid resolution. The numerical solution is within 0.5% of the exact
solution at the 5050 resolution. Fig. (2.1b) plots the error in radius between the
exact and numerical solutions at t = 0:15 for several grid resolutions. For consistency,
the time step was kept at the same value of t = 3 (10,5 ) for each of the resolutions
in this gure. Consistent with the study by LeVeque and Li [136] the front tracking
method exhibits between linear and quadratic convergence. Note that the numerical
results are in good agreement with the exact solution even at low resolutions. Thus
even though the convergence is not quadratic, the constant in the error estimate is
small.
28
The Stefan number varies with the temperature drop that the material must undergo, but for water and some nonmetals the Stefan number is typically less than
unity, while for metals the Stefan number is generally 1-10. For high Stefan number the latent heat released during solidication is small and the problem becomes
essentially one of pure conduction without phase change. For low Stefan number,
the latent heat released is high and the solidication problem is dominated by conduction of the latent heat liberated at the interface. Fig. (2.2) shows numerical and
exact results for large and small Stefan numbers. A plot of average radius vs. time
is shown for St = 10 and St = 0:1 for a 5050 grid resolution with a heat sink
strength of QL = 10. Agreement with the exact solution is excellent for St = 10. For
St = 0:1 the higher latent heat released results in a larger heat ux discontinuity at
the interface. This situation is more dicult to handle numerically but the results
are still within 2% of the exact solution.
In Fig. (2.3) the average radius vs. time is plotted for two cases where the
material properties in the liquid and solid phases are unequal. The results for a
liquid to solid thermal conductivity ratio, Kl=Ks = 0:2 and a liquid to solid specic
heat ratio, cl=cs = 2 agree well with the corresponding exact solutions despite the
discontinuities in the material properties. For comparison, water has Kl =Ks 0:25
and cl=cs 2. Here, St = 1, the heat sink strength is increased to QL = 50 and
the grid resolution is increased to 100100. Temperature proles corresponding to
the case where Kl=Ks = 0:2 are plotted along the X -axis at a one-dimensional slice
through Y = 0:51 in Fig. (2.4). The four curves are plotted for increasing time from
left to right at time increments of 0.02. The large temperature gradients near the line
heat sink at X = 0:505 (r = 0) and the discontinuity in temperature gradient at the
melting temperature, T = 0, due to the release of latent heat and the unequal thermal
29
conductivities are reproduced accurately by the numerical results. The liquid/solid
interfaces for the curves in Fig. (2.4) are plotted in Fig. (2.5). As the interface
expands outward in concentric circles more points are added to the interface in order
to maintain an interface grid resolution of 0:4 < d=h < 1:6.
Fig. (2.6) demonstrates the ability of the method to handle topology changes. In
this case four heat sinks create four expanding circular regions of solid material. As
these regions approach each other, their interfaces are ruptured when the distance
between any interface points is less than an arbitrarily chosen proximity of p = 0:005.
The four solid regions merge to form a large solid region with an entrapped liquid
region. This four cusped liquid region circularizes and eventually disappears as it
completely solidies.
2.4.2 Unstable Solidication
Results for unstable solidication into an undercooled liquid are now presented.
For all of the simulations the domain boundaries are insulated. The problem is
solved on the entire domain and no symmetry requirements on the interface shape are
imposed. The primary physical parameters are the dimensionless undercooling, St,
the capillary parameter, , and the ratios of specic heat and thermal conductivity in
the solid and liquid regions. The process of solidication into an undercooled liquid
is inherently unstable and sensitive to initial conditions. The degree of instability
in the growing solid depends on the choice of physical parameters. Typically, for
high and low St the liquid/solid interface remains smooth for long periods of
growth while for low and high St the interface deforms quickly into a branching
dendritic pattern. The eects of dierent physical and numerical parameters on the
time evolving interface shapes as well as on global quantities of interest such as solid
30
fraction and dendrite arm length are examined.
Fig. (2.7) shows results for four dierent grid resolutions. For these cases St =
,0:5, = 0:002 and # = 0:002 with no anisotropy (s = k = 0) and equal material
properties in the liquid and solid. Interface points are added or deleted as required
to maintain a point spacing of 0:4 < d=h < 1:6. The computational domain is a
square with sides of length 4. The shape of the initial interface (xf ; yf ) is specied
by
xf = xc + R cos();
yf = yc + R sin()
(2.29)
where is measured counterclockwise from the x,axis and
R = Ro + Rb cos [M ( + o)] :
(2.30)
In this gure, the choice (xc; yc) = (2; 2), Ro = 0:1, Rb = 0:02, M = 4 and o = 0
produces a perturbed circle with four lobes aligned with the coordinate axes. The
time-evolving interfaces are plotted for all four grid resolutions at equal nondimensional time increments of 0.03. The nal interface plotted for the 100100 mesh
contains 1,345 interface points and is at time 0.09. The 200200 mesh contains
1,746 points and is at time 0.63. For the 300300 mesh the last shape contains
2,053 points at time 0.81 and for 400400, 2,542 points also at time 0.81. Note that
these results show the eect of time as well as grid resolution since the maximum
time step decreases with grid resolution. Fig. (2.8) shows plots of various interface
quantities for the increasing grid resolutions shown in Fig. (2.7). Plotted vs. time
are the fraction of material solidied and the maximum radius (maximum distance
of an interface point from the centroid of the initial interface, (xc; yc) = (2; 2)).
The grid resolution obviously has considerable eect on the solution. For the lowest resolution, the interface branches excessively, and a comparison with the results
31
on the ner grids shows that the solution is dominated by grid eects. The solution
of the ner grids, on the other hand, all have essentially the same features. The four
initial protrusions, or ngers, grow and become wider, then split. The eight resulting
ngers also become wider as they grow and eventually split. The dierence between
the solutions on the three grids is the time when the ngers split. On the coarsest
grid (200200) the splitting takes place early and there are sixteen well developed
ngers at the nal time. As the grid is made ner, the ngers split later, although
the dierence between the 300300 grid and the 400400 grid is relatively small.
The results are very much as expected. The ngers split due to secondary instabilities and these depend sensitively on the amount of noise present. Generally, larger
perturbations on the coarser grid and an earlier onset of instability are expected.
Therefore, while the solutions in Figs. (2.7) and (2.8) are not fully converged in the
sense that the two nest grids give completely identical solutions, it appears that the
physical phenomena are resolved for both grids. Or, said dierently, the important
distinction between the solution on the 100100 grid and the three ner grids is that
the instabilities on the 100100 grid are completely articially created by the grid
noise while on the ner resolutions the grid noise only acts to trigger the onset of
the physical Mullins-Sekerka instability. As the resolution increases and the noise
decreases, this physical instability is triggered at later times. In a physical situation,
the Mullins-Sekerka instability would be similarly triggered by thermal uctuations
due to uid convection or random statistical uctuations. In most cases, one would
want to trigger these instabilities anyway, so perturbations induced by the grid are
not necessarily undesirable. In order to completely eliminate the grid eect, noise at
a suciently high level could be introduced to swamp the grid noise.
A consequence of the earlier tip splitting (and smaller tip radii) produced on
32
coarser grids is an increase in tip growth speed. The increased growth speeds at
lower grid resolutions is evident in Fig. (2.8), particularly in the plot of maximum
radius vs. time. This trend of increased tip speed with decrease in tip radius is an
expected result. Indeed, the trend is similar to Ivantsov's simplied analysis of a
growing dendrite [99]. There he found an inverse relationship between dendrite tip
speed, V , and tip radius, R, namely, V R = const.
Although the results are not shown here, the inuence of interface point spacing,
d has also been studied. The same computations as shown in Fig. (2.7) have been run
with interface point spacings of 0:2 < d=h < 0:8 and 0:8 < d=h < 3:2. For the three
higher grid resolutions, the eect on the interface shape as well as the solid fraction
and maximum radius is negligible. The results appear to be independent of interface
point spacing for a reasonable choice of point spacings. This also indicates that
the interface curvature calculation (see appendix A.1) using a simple fourth-order
polynomial curve t is suciently accurate. All of the remaining computations shown
here have a point spacing of 0:4 < d=h < 1:6.
To demonstrate that the solution is independent of grid orientation, runs with
dierent initial interface rotations, o, are compared. The interface in Fig. (2.7d)
with o = 0 was compared to two runs with o = 27 and o = 45. The interface
shapes from the two runs at t = 0:21 were then rotated back 27 and 45 degrees
respectively and plotted over each other for comparison. The 3 interfaces were nearly
indistinguishable from one another. A plot of this result would look identical to the
interface in Fig. (2.7d) at t = 0:21 and thus is not included here.
In Fig. (2.9a) an initially ve-fold symmetric interface (M = 5) maintains its
symmetry throughout the computation. The last interface shown contains 2,113
points at t = 0:42. Other than M = 5, the parameters for this run are the same
33
as in Fig. (2.7d). Clearly there is no grid induced anisotropy and the alignment of
the tracked front with the underlying stationary grid does not aect the solution. A
plot of the interface curvature as a function of the surface distance in Fig. (2.9b)
demonstrates the ve-fold symmetric structure. This plot corresponds to the last
interface in Fig. (2.9a). The interface curvatures for each of the ve main branches
in Fig. (2.9a) are plotted over each other. The curvature repeats nearly identically
ve times around the interface. The slight discrepancies in curvature from one branch
to another may be partly attributed to the eect of the domain boundaries. The
smoothness of the curvature plot again suggests that the simple fourth-order curve
t used in the calculation of curvature (see appendix A.1) is suciently accurate.
In homogeneous nucleation a small particle of solid forms and grows within its
own liquid due to statistical variations in the distribution of clusters of atoms within
the liquid. The total variation in free energy due to the formation of a small spherical
particle of solid of radius, R, has volume and surface contributions
G = , 43 R3 TLo T + 4R2 :
m
(2.31)
The condition for homogeneous nucleation is that the total variation in free energy,
G, of the nucleus is maximum. Thus the critical nucleation radius, R, above
which a solidifying particle will grow and below which it will collapse is dened for
the condition @ G=@R = 0. In a two-dimensional system this critical radius is
:
R = , St
(2.32)
In order to see if this critical radius could be simulated three calculations all with
= ,St = 0:5 and thus a theoretical critical radius of R = 1 were performed. The
initial interface for the rst case is a circle of radius 1.01, slightly larger than the
critical radius. The second case started with a circle of radius 1 and the third with
34
a radius of 0.99. Fig. (2.10) plots the average radius vs. time for all three interfaces.
The circle of initial radius 1.01 grows, that with initial radius of 0.99 collapses.
The circle that started at the critical radius R = 1 stays at 1. As predicted, the
simulations show that a nucleus below the critical radius will not grow whereas a
nucleus above the critical radius does grow. These simulations were run with equal
material properties in the liquid and solid, isotropic surface tension and no kinetic
eects, # = 0. The grid resolution was 300300 in a 1010 square domain.
Next the eect of discontinuous thermal conductivity and specic heat between
liquid and solid phases is investigated. The results for the interface evolution for
various thermal conductivity and specic heat ratios are shown in Figs. (2.11) and
(2.12). (Figs. (2.11b) and (2.12b) are the same as Fig. (2.7c) and are shown again
here to facilitate comparisons.) The parameters for these cases are the same as in
Fig. (2.7c) except that the material property ratios are varied. Again, the interfaces
are plotted at equal nondimensional time increments of 0.03.
Fig. (2.11) shows the eect of increasing Kl =Ks with cl=cs = 1. Increasing the
thermal conductivity ratio primarily aects the growth rate of the solid and has little
eect on the shape or stability of the interface. The relationship between interface
velocity and thermal conductivity ratio becomes apparent upon examination of the
nondimensional form of the Stefan condition, Eq. (2.6)
Kl rT n :
1 + ccl , 1 Tf V = rTs , K
l
s
s
(2.33)
Compared to the temperature gradient on the liquid side of the interface, the temperature gradient on the solid side remains small throughout the computation since
both the interface temperature and the temperature in the bulk of the solid remain
near the equilibrium freezing temperature. If the temperature gradient in the solid
35
is neglected and for equal specic heats, it is clear that the interface velocity is
proportional to the thermal conductivity ratio.
Kl rT n :
V = ,K
l
s
(2.34)
The results in Figs. (2.11a-c) are consistent with this analysis. As the liquid to solid
thermal conductivity ratio increases from Kl=Ks = 0:2 to 2, a ten-fold increase, the
growth rate of the solid increases roughly three-fold. A comparison with the exact
solution of the problem of stable solidication by a line heat sink discussed earlier
shows that in the stable case the magnitude of the growth rate roughly doubles with
the same ten-fold increase in Kl=Ks . (Note that the trend of increased growth rate
with increased Kl =Ks as given by Eqs. (2.26) and (2.27) is reversed for the problem
of stable solidication. This is due to the fact that the liquid is not undercooled and
the temperature gradients in the liquid and solid regions are both positive.)
Fig. (2.12) shows the eect of increasing cl=cs with Kl=Ks = 1. Increasing the
specic heat ratio even slightly has a dramatic eect on the interface stability as well
as the growth rate. The eect is much more pronounced than it is for a change in
the thermal conductivity ratio discussed above. A higher liquid to solid specic heat
ratio produces a fast growing unstable solid while a low ratio produces a slow growing
more stable solid shape. In going from cl=cs = 0:83 to 1:2 the growth rate nearly
doubles. In comparison with the problem of stable solidication by a line heat sink,
such a large change in growth rate with a small change in cl=cs is not expected. The
exact solution for the stable problem shows only a 10 % change in the magnitude
of the solid growth rate for the same change in cl=cs . (Again, note that the trend
of increased growth rate with increased cl=cs as given by Eqs. (2.26) and (2.27) is
reversed for the problem of stable solidication due to the fact that the liquid is not
36
undercooled and the temperature gradients in the liquid and solid regions are both
positive.)
There appear to be two mechanisms related to the specic heat ratio which
explain these eects. The rst mechanism is simply a manifestation of elementary
unsteady heat conduction and does not take interface curvature into account. The
overall increase in growth rate with increased specic heat ratio is a direct result
of the varying ability of materials with dierent specic heats to adjust to thermal
changes. A material with a low volumetric specic heat will respond quickly to
thermal changes while one with a high specic heat will respond slowly. Therefore,
a liquid with a high specic heat has less of an ability to diuse the latent heat
released in a given time. This results in a steeper temperature gradient in the liquid
adjacent to the interface. According to the Stefan condition, Eq. (2.33), a larger
liquid side temperature gradient results in a higher interface velocity. Furthermore,
interface movement in the direction of the liquid steepens the liquid side temperature
gradient even more. Conversely, a lower interface velocity would result from a lower
liquid phase specic heat. The second mechanism produces an increase in interface
instability with increased cl=cs and can be explained using Eqs. (2.33) and (2.15). If
the second order temperature term, anisotropy and interface kinetics in Eq. (2.15)
are neglected and the two equations are combined to eliminate Tf then for equal
thermal conductivities Eq. (2.33) can be written as
1 , ccl , 1 V = [rTs , rTl] n :
s
(2.35)
The coecient of the local normal interface velocity, V , depends on the specic
heat ratio and the local interface curvature. If cl=cs > 1 then in regions of positive
curvature, where the interface grows the fastest, this coecient is less than 1. For a
37
given q, V must increase to compensate for the low coecient. In regions of negative
curvature, the coecient of V is greater than 1 and thus V is lower for the same
q. The net result is an increase in interface instability. Regions of the interface
where the interface curvature is positive grow faster and regions of the interface
where the curvature is negative slow down. The opposite occurs if cl=cs < 1. In
regions of positive curvature, the coecient of V is greater than 1. For the same q,
V must be lower. In regions of negative curvature, the coecient of V is less than 1
and thus V is higher for the same q. The net result is a more stable interface where
positive curvature regions grow relatively slowly and negative curvature regions grow
relatively rapidly. Furthermore, the two mechanisms described above are coupled in
the sense that the local instabilities produced by the second mechanism are enhanced
by the global velocity changes in the rst.
The above argument and the results in Fig. (2.12) suggest that discontinuity
of specic heat is a crucial consideration when modeling solid growth rates and
interface morphology in the dendritic freezing of water, (cl=cs 2), and some metals such as lead (cl=cs 1:1), copper and nickel (cl=cs 1:05) and molybdenum
(cl=cs 1:7) [137]. Most common pure metals have cl=cs 1 and thus tend toward
more instability. However, it would be dicult to experimentally isolate the eect
of discontinuous specic heat, for example, on dierent materials. Other physical
parameters such as surface tension would usually vary along with the specic heat
ratio.
The strong interface branching exhibited in Figs. (2.12c,d) may indicate that
these results are not completely converged. However, the results of a resolution study
show about the same rate of convergence as in Fig. (2.7). As the grid resolution
is increased the interfaces become more symmetric and the interface growth rate is
38
slightly lower. Thus the primary trends of increased interface velocity and instability
remain and are due to the increase in cl=cs .
A plot of solid fraction versus time in Fig. (2.13) clearly shows the increase in
growth rate with increased cl=cs . Also plotted as dashed lines are the maximum theoretical solid fractions for the various specic heat ratios. The maximum theoretical
amount of solid that can be formed from an undercooled liquid in an insulated cavity
can be found from simple energy conservation to be
fs = , ccl St
s
(2.36)
which states that the only source of energy available to raise the temperature of the
liquid to the equilibrium melting temperature comes from the latent heat released
at the interface. The numerically calculated value of the solid fraction at large time
agrees well with the theoretical limit but begins to dier slightly at higher cl=cs.
This discrepancy is due to the higher interface curvatures at higher cl=cs . At high
curvatures, interface temperatures deviate from the equilibrium freezing temperature
because of capillary eects (the Gibbs-Thomson condition, Eq. (2.15)). Eq. (2.36)
does not take these capillary eects on the temperature into account.
In Fig. (2.14) the eect of a six-fold anisotropy (ns = 6) in the surface tension
is shown. The interfaces are plotted at equal increments of t = 0:003 and the last
interface shown is at t = 0:03 with 5,854 points. The anisotropy strength, s, is
0.4 and the other parameters remain the same as in Fig. (2.7d). The initial fourlobed interface no longer imparts a four-fold symmetry to the evolving interface as
it did with the isotropic simulations. Growth occurs along six primary directions
dictated by the surface tension anisotropy. The undercooling has a high value of
St = ,0:8 which produces a more complex shape with smaller features. In order to
39
spatially resolve this computation it is necessary to be able to resolve the smallest
features with at least as many mesh points as were required by the smallest features
in Fig. (2.7d). The smallest features of Fig. (2.7d) are approximately 0.08 across
and are resolved on a 400400 grid. The smallest features in Fig. (2.14) are the
developing side branches which are about 0.06 across and thus should be resolved on
the 800800 grid used here. The six primary dendrite arms are certainly resolved
at this scale. Although the major structures grow symmetrically, the small side
branch disturbances emanating from the primary arms begin to grow asymmetrically.
Murray et al. [25] have found in their phase eld computations that the production
of side branches was very sensitive to noise whereas the main dendrite tips were
insensitive to noise. Fig. (2.14) agrees with this assessment. The main branches
grow symmetrically with a constant speed, after the initial transient, of V = 40 (see
Fig. (2.16)) and tip curvature of approximately = 60 whereas the side branches
develop asymmetrically and are sensitive to small numerical inaccuracies and grid
noise. Once the symmetry is broken, however slightly, the competition between
neighboring side branches results in the growth of some branches and melting back
of others.
A relation for the shortest wavelength at which a disturbance on a planar interface, moving with velocity V , becomes unstable is given by Mullins and Sekerka [94]
s V :
r
(2.37)
Although the interface in the simulations is not planar, it can be assumed that the
same relationship approximately holds. It is clear that the problem of resolving the
shorter unstable wavelengths at low or high undercooling and thus high V , limits
all grid based numerical simulations.
40
In Fig. (2.15), a more developed side branch structure results from lower surface
tension. The parameters here are the same as in Fig. (2.14) except the capillary
parameter is reduced to = 0:001. The initial temperature eld for this computation
is a uniform temperature given by T (x; t = 0) = St and not by Eq. (2.20) as in the
previous runs. The interfaces are plotted at equal increments of t = 0:003 and the
last interface shown is at t = 0:021 with 11,002 points. With reduced by half, the
primary dendrite arms grow with a larger but still constant speed of V = 60 (see
Fig. (2.16)) and a higher tip curvature of approximately = 100. Consistent with
Eq. (2.37), the shortest wavelength of the side branch instabilities in Fig. (2.15)
is roughly 3/5 of what it is in Fig. (2.14). With these smaller features the spatial
resolution limit is approached for the 800800 grid used here. The competition
between neighboring side branches is more intense in this gure and results in an
earlier onset of asymmetry.
For the two cases in Figs. (2.14) and (2.15), Fig. (2.16) plots the dendrite length
(length of the primary dendrite arm in the rst quadrant) vs. time. The slopes
are constant throughout most of the computation which indicates that the dendrite
tip speed, V , is constant. For = 0:001, V is approximately 60 while for a higher
surface tension, = 0:002, the tip speed drops to approximately 40.
Fig. (2.17) demonstrates the ability of the front tracking method to easily handle
qualitatively realistic topology changes. The interfaces are allowed to merge and
form islands of trapped liquid. The parameters for this case are the same as in Fig.
(2.15). Interfaces are merged and reconnected when the distance between any two
points is less than p = 0:013. The interface consists of eleven surfaces, 10,890 points
and is plotted at t = 0:021. Here, the phenomena of liquid trapping and coarsening
are reproduced. Several of the liquid islands have circularized due to the eect of
41
surface tension. The smaller islands eventually completely solidify. Comparison
between this gure and Fig. (2.15) shows that, other than the topological changes,
the general structure of the interface remains the same.
The breakdown of a planar interface in unstable solidication is simulated in Fig.
(2.18). Here the parameters are the same as in Fig. (2.15) except that the domain
is now periodic in the x,direction and the dendrites grow upward from a planar
interface. The initial interface is given a very small random perturbation to trigger
the onset of the instability. The nal interface shape consists of 10,161 points. The
competition among the growing perturbations is clearly shown. Only eleven primary
dendrite arms survive while the smaller ones melt back slightly. Secondary arms
form on the sides of some of these primary dendrites but only when the spacing
between the primary arms is large enough. When the primary arm spacing is small
the growth of secondary side branches is suppressed.
In Fig. (2.19) a simulation of the growth of a four-fold symmetric (ns = 4)
succinonitrile dendrite is attempted. Agreement with experimental results is not
expected since the simulation is for a two-dimensional domain with insulated boundaries. The simulation is intended to demonstrate that qualitatively realistic results
can be achieved using realistic values of the physical parameters. If a length scale of
l = 0:01 cm is chosen then the thermal properties of succinonitrile [138] correspond
to = 2:8 (10,5 ), # = 2:46 (10,4 ), cl=cs = Kl=Ks 1 with a negligible density
change upon solidication. The undercooling St = ,0:1 corresponds to ,2:31 K .
The magnitude of the capillary anisotropy is s = 0:08 which is in the range of values
reported in [139] and [140]. There is no anisotropy in the kinetic mobility. Note that
the form of the anisotropic capillary parameter used in the above references is
n() = [1 , s cos (4)]
(2.38)
42
which is slightly dierent than Eq. (2.16). The interfaces in Fig. (2.19) are plotted at
equal increments of t = 0:2 and the last interface shown is at t = 4:6. The growth of
the four arms in Fig. (2.19) is symmetric and the features are well resolved. One can
see the beginnings of side branch development, however for such a low undercooling
the eect of the insulated walls is felt early in the computation and slows the growth
dramatically.
2.5 Conclusions
A new front tracking method based on standard nite dierence techniques is
presented for stable and unstable solidication of pure substances. Using a simple
iterative algorithm, The governing phenomenological equations and interface conditions are directly solved and the introduction of additional non-physical simulation
parameters is thereby avoided. New features of the method include concepts from
the Immersed Boundary Technique [141] to transfer information between the moving
front and the stationary nite dierence grid and the construction of an indicator
function which enables computations with discontinuous material properties. The
method is also easily able to simulate surface tension and kinetic mobility anisotropies
as well as topology changes.
It appears that the method performs well when compared to exact solutions for
stable solidication by a line heat sink. Grid resolution studies for computations of
dendritic structures indicate that the method converges well under grid renement.
Although the physical features are well resolved, the eect of larger grid noise at lower
resolutions acts to trigger the physical instabilities at earlier times. The results are
shown to be free of any grid induced anisotropy and grid orientation eects. As a test
of thermodynamic consistency, the critical nucleation radius is simulated to within
43
1% and the maximum theoretical solid fraction for solidication in an insulated
cavity is matched. For more unstable parameters, in particular for high St, low
or high cl=cs , the interface complexity and thus the required resolution increases.
Although the method is thus eventually limited by resolution for highly unstable sets
of parameters, converged results have been achieved for realistic values. The results
in this work are limited to two-dimensions, however, and are not directly comparable
to three-dimensional experimental results.
For unstable, dendritic solidication with discontinuous material properties, it is
found that the interface grows faster with an increase in the liquid to solid thermal
conductivity and volumetric specic heat ratios. In addition, with a small increase in
the specic heat ratio, the solid grows faster and the interface becomes more unstable
than would be expected by an analysis of similar conditions in stable solidication.
Two mechanisms which contribute to and magnify this instability are identied.
The results indicate that in modeling growth rates and interface morphology in the
dendritic solidication of water and many pure metals, the discontinuity of material
properties between liquid and solid phases is an important consideration. It appears
that similar computations which take into account unequal thermal conductivity and
specic heat have not previously been reported.
The main objective of the present work was to develop and verify the numerical
method. In future work, the capability of the method will be extended to perform
three-dimensional computations of dendritic solidication and thereby allow comparisons with experimental results. The main diculties in three-dimensional simulations are the computational requirements and the more complex surface-tracking
algorithms. However, such surface-tracking has successfully been implemented in
[64{66], for example. Due to the high computational expense the method will most
44
likely need to incorporate adaptive mesh routines which allow ner resolution near
the interface and also second-order time integration for increased accuracy in long
time simulations.
45
0.5
Radius
0.4
0.3
10 x 10
20 x 20
50 x 50
Exact
0.2
0.1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time
(a) Average Radius vs. Time
−2
10
Numerical Results
log(error)
Li
ne
ar
Co
Qu
nv
erg
atic
adr
en
ce
nce
rge
nve
Co
−3
10
1
2
10
10
log(1/h)
(b) Convergence with Grid Resolution
Figure 2.1: Stable two-dimensional solidication by a line heat sink. A grid resolution test. In (a) the average radius of all the interface points vs. time is
compared to the exact solution for three dierent grid resolutions. In (b)
the error in radius at t = 0:15 between the exact and numerical solutions
is plotted for several grid resolutions. As expected, the front tracking
method exhibits between linear and quadratic convergence. St = 1,
QL = 10, Kl =Ks = cl=cs = 1.
46
0.5
0.45
St=10
0.4
St=0.1
Radius
0.35
0.3
0.25
Numerical
Exact
0.2
0.15
0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Figure 2.2: Eect of varying the Stefan number, St. The average radius of all the interface points vs. time is compared to the exact solution for two dierent
Stefan numbers. 5050 grid, QL = 10, Kl=Ks = cl=cs = 1.
47
0.5
0.45
0.4
k l / ks = 0.2
Radius
0.35
c l / cs = 2
0.3
0.25
0.2
Numerical
Exact
0.15
0.1
0
0.01
0.02
0.03
0.04
0.05
Time
0.06
0.07
0.08
0.09
0.1
Figure 2.3: Eect of varying material properties. The average radius of all the interface points vs. time is compared to the exact solution for a thermal
conductivity ratio Kl=Ks = 0:2 and a specic heat ratio of cl=cs = 2.
100100 grid, St = 1, QL = 50.
48
0
-5
Temperature
-10
-15
-20
Numerical
Exact
-25
-30
0.55
0.6
0.65
0.7
0.75
X
0.8
0.85
0.9
0.95
1
Figure 2.4: Temperature proles along the X -axis at Y = 0:51. The four curves
are for increasing time from left to right: t = 0:003, 0:023, 0:043, 0:063.
Kl =Ks = 0:2, cl=cs = 1, 100100 grid, St = 1, QL = 50.
49
Numerical
Exact
1
0.8
Y
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
X
Figure 2.5: Liquid/solid interfaces for the four curves in Fig. (2.4). As the interface
expands outward in concentric circles more points are added to the interface in order to maintain an interface grid resolution on the order of
a stationary grid mesh block, h. Kl =Ks = 0:2, cl=cs = 1, 100100 grid,
St = 1, QL = 50.
50
1.4
1.3
1.2
Y
1.1
1
0.9
0.8
0.7
0.6
0.6
0.7
0.8
0.9
1
X
1.1
1.2
1.3
1.4
Figure 2.6: Topology change. Four heat sinks create four expanding regions of
solid. As the solidifying regions approach each other their interfaces
are ruptured when the distance between any interface points is less than
p = 0:005. The four solid regions merge to form a large solid region with
an entrapped liquid region in the center. This four cusped liquid region
circularizes and eventually disappears as it completely solidies. The
physical domain is 22, 100100 grid, St = 1 and Kl=Ks = cl=cs = 1
4
4
3
3
Y
Y
51
2
1
0
0
2
1
1
2
X
3
0
0
4
1
4
4
3
3
2
1
0
0
3
4
3
4
(b) 200 x 200
Y
Y
(a) 100 x 100
2
X
2
1
1
2
X
(c) 300 x 300
3
4
0
0
1
2
X
(d) 400 x 400
Figure 2.7: A grid resolution study for dendritic solidication in an insulated cavity. The interface evolution for four grid resolutions is plotted at equal
nondimensional time increments of 0.03. As the grid resolution increases
the interface shapes become more symmetric and grid independent. The
results reproduce the classic tip splitting instability of dendritic solidication. St = ,0:5, = 0:002, # = 0:002 with no anisotropy (s = k = 0)
and equal material properties (cl=cs = Kl=Ks = 1).
52
0.5
0
0.45
100 x 100
200 x 200
0x
30
30
400 x 400
0.4
Solid Fraction
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
Time
0.8
1
1.2
1
1.2
3
200 x 200
100 x 100
300 x 300
2
Maximum Radius
400 x 400
1
0
0
0.2
0.4
0.6
Time
0.8
Figure 2.8: The solid fraction and maximum radius vs. time are plotted for the
various grid resolutions corresponding to the interface plots in Fig. (2.7).
The solid fraction and maximum radius demonstrate better than linear
convergence with increasing grid resolution. St = ,0:5, = 0:002,
# = 0:002 with no anisotropy (s = k = 0) and equal material properties
(cl=cs = Kl=Ks = 1).
53
4
Y
3
2
1
0
0
1
2
X
3
4
(a) Interface Evolution
6
4
2
Interface Curvature
0
−2
−4
Branch 1
−6
2
−8
3
4
−10
−12
0
5
0.1
0.2
0.3
0.4
0.5
0.6
Surface Distance
0.7
0.8
0.9
1
(b) Interface Curvature vs. Surface Distance
Figure 2.9: At top an initially ve-fold symmetric interface (M = 5) maintains
its symmetry throughout the computation. There is no grid induced
anisotropy. Below, a plot of the interface curvature as a function of the
surface distance for the last interface in (a) shows that the curvature
repeats nearly identically ve times around the interface. The interface
curvatures for each of the ve main branches in (a) are plotted over
each other. M = 5, St = ,0:5, = 0:002, # = 0:002, s = k = 0,
cl=cs = Kl=Ks = 1 and a 400400 grid.
54
1.25
1.2
1.15
Initial Radius=1.01R*
1.1
Radius
1.05
Initial Radius=R*
1
0.95
0.9
Initial Radius=0.99R*
0.85
0.8
0.75
0
1
2
Time
Figure 2.10: Simulation of the critical nucleation radius. In a two-dimensional system the critical nucleation radius is R = ,=St. Here, = ,St = 0:5
and thus the critical radius is R = 1. The radius vs. time for three
dierent initial radii is shown. A circle of initial radius 1.01 grows,
one with initial radius of 0.99 collapses. A circle started at the critical
radius R = 1 stays at 1. As predicted, the simulations show that a
nucleus below the critical radius will not grow whereas a nucleus above
the critical radius does grow. # = 0, no anisotropy (s = k = 0) and
cl=cs = Kl=Ks = 1. The grid resolution is 300300 in a 1010 square
domain.
4
4
3
3
Y
Y
55
2
1
0
0
2
1
1
2
X
(a)
3
Kl =Ks
0
0
4
1
=02
2
X
(b)
:
Kl =Ks
3
4
=10
:
4
Y
3
2
1
0
0
1
2
X
(c)
Kl =Ks
3
4
=20
:
Figure 2.11: Eect of varying the liquid to solid thermal conductivity ratio. The
interfaces are plotted at equal time increments of 0.03. (Note that Fig.
(2.11b) is the same as Fig. (2.7c).) Increasing the thermal conductivity
ratio increases the growth rate of the solid but has little eect on the
interface shape or stability. St = ,0:5, = 0:002, # = 0:002 with no
anisotropy (s = k = 0), cl=cs = 1. and a 300300 grid.
4
4
3
3
Y
Y
56
2
1
0
0
1
1
2
X
(a)
cl =cs
3
0
0
4
2
X
(b)
:
4
4
3
3
2
1
0
0
1
= 0 83
Y
Y
2
cl =cs
3
4
3
4
=10
:
2
1
1
2
X
(c)
cl =cs
3
=12
:
4
0
0
1
2
X
(d)
cl =cs
=14
:
Figure 2.12: Eect of varying the liquid to solid specic heat ratio. The interfaces
are plotted at equal time increments of 0.03. (Note that Fig. (2.12b)
is the same as Fig. (2.7c).) Increasing the specic heat ratio increases
the instability of the interface as well as the growth rate. St = ,0:5,
= 0:002, # = 0:002 with no anisotropy (s = k = 0), Kl=Ks = 1. and
a 300300 grid.
57
0.8
c l / cs = 1.4
0.7
1.2
0.6
1.0
Solid Fraction
0.5
_
0.83
0.4
0.3
0.2
0.1
0
0
Numerical
Theoretical Limit
0.5
1
1.5
Time
Figure 2.13: The solid fraction vs. time shows the increase in growth rate of the solid
with increasing cl=cs. Also plotted are the maximum theoretical solid
fractions for the various specic heat ratios. The numerically calculated
values of the solid fraction at large time agree well with the theoretical
limits but begin to dier slightly at higher cl=cs. This discrepancy is due
to capillary eects on the temperature which are not taken into account
in the theoretical solid fraction. St = ,0:5, = 0:002, # = 0:002 with
no anisotropy (s = k = 0) and a 300300 grid.
58
4
Y
3
2
1
0
0
1
2
X
3
4
Figure 2.14: With six-fold anisotropy (ns = 6) in the surface tension, growth occurs
along six primary directions. The interfaces are plotted at equal increments of t = 0:003 and the last interface shown is at t = 0:03 with
5,854 points. The undercooling has a high value of St = ,0:8 which
produces a more complex shape with smaller features. 800800 grid,
= 0:002, s = 0:4 # = 0:002, k = 0 with equal material properties
(cl=cs = Kl=Ks = 1).
59
4
3.5
3
Y
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
X
2.5
3
3.5
4
Figure 2.15: A more well developed side branch structure results from lower surface
tension. The parameters here are the same as in Fig. (2.14) except
the capillary parameter is reduced to = 0:001. The interfaces are
plotted at equal increments of t = 0:003 and the last interface shown
is at t = 0:021 with 11,002 points. 800800 grid, = 0:001, s = 0:4
# = 0:002, k = 0 with equal material properties (cl=cs = Kl=Ks = 1).
60
1.8
1.6
1.4
Dendrite Length
1.2
1
0.8
0.6
σ =0.001
σ =0.002
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Time
Figure 2.16: A plot of the dendrite length (length of the primary dendrite arm in
the rst quadrant) vs. time for the two cases in Figs. (2.14) and (2.15).
The slopes are constant throughout most of the computation which
indicates that the dendrite tip speed, V , is constant. For = 0:001, V
is approximately 60 while for a higher surface tension, = 0:002, the
tip speed drops to approximately 40.
61
4
3.5
3
Y
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
X
2.5
3
3.5
4
Figure 2.17: Demonstration of topology change with six-fold anisotropy (ns = 6) in
the surface tension. Interfaces are merged when the distance between
any two points is less than p = 0:013. The interface consists of eleven
surfaces, 10,890 points and is plotted at t = 0:021. The qualitatively
realistic phenomena of liquid trapping and coarsening are reproduced.
Several of the liquid islands have become circularized due to the eect of
surface tension. The smaller islands eventually completely solidify. The
parameters for this case are the same as in Fig. (2.15). Comparison between this gure and Fig. (2.15) shows that, other than the topological
changes, the general structure of the interface remains the same.
62
4
3.5
3
Y
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
X
2.5
3
3.5
4
Figure 2.18: Simulation of the breakdown of a planar interface in unstable solidication. The parameters are the same as in Fig. (2.15) except that
the domain is now periodic in the x,direction and the dendrites grow
upward from a planar interface. The initial interface is given a very
small random perturbation to trigger the onset of the instability. The
nal interface shape consists of 10,161 points. The competition among
the growing perturbations is clearly shown. Only eleven primary dendrite arms survive while the smaller ones melt back slightly. Secondary
arms form on the sides of some of these primary dendrites but only
when the spacing between the primary arms is large enough. When the
primary arm spacing is small the growth of secondary side branches is
suppressed.
63
4
Y
3
2
1
0
0
1
2
X
3
4
Figure 2.19: Growth of a two-dimensional succinonitrile dendrite. At a length scale
of l = 0:01 cm, the thermal properties of succinonitrile [138] correspond
to = 2:8 (10,5 ), # = 2:46 (10,4 ), cl=cs = Kl =Ks 1. The undercooling St = ,0:1 corresponds to ,2:31 K . The growth of the four
arms is symmetric and the features are well resolved. One can see the
beginnings of side branch development, however for such a low undercooling the eect of the insulated walls is felt early in the computation
and slows the growth dramatically. s = 0:08, ns = 4, k = 0.
CHAPTER III
ALLOY SOLIDIFICATION
3.1 Introduction
Nearly all materials of engineering interest have at some point solidied from
a liquid state. The exact nature of the solidication process determines the microstructure and thus the physical properties of the solid material. For example,
the electrical properties of semiconductors and the mechanical properties of superalloy turbine blades are inuenced by crystallographic defects and chemical inhomogeneities formed during solidication. Advanced manufacturing processes involving
rapid solidication and spray casting for the production of net or near-net shaped
components depend on the ability to control and understand the processing conditions during solidication [142]. Directional solidication is one of several commercially important methods for solidifying alloys [143]. The basic idea in this process
is to translate the part to be solidied from the hot to the cold end of a furnace.
The translation velocity, Vo , and the temperature gradient in the solidifying zone
of the furnace, G, along with the initial alloy composition, Co, are the controllable
process variables which determine the microstructure and the physical properties of
the nal solidied part. The liquid-solid interface during solidication is generally
not planar but forms a cellular or dendritic structure depending on the processing
64
65
conditions. The formation of chemical nonuniformities which become frozen into the
solid is closely coupled to the dynamics of the interface. Understanding the behavior
and geometry of the interface and linking this behavior to the processing conditions
is a fundamental goal of metallurgists and materials scientists.
Until recently this understanding was mostly empirical. In the 1950's, Chalmers
and co-workers [144], [145] introduced the constitutional supercooling criterion for
the onset of instability in a planar interface. The now classic explanation of this instability mechanism is that the solute rejected into the liquid ahead of the advancing
liquid-solid interface (in this case the partition coecient, k, see Eq. (3.3) is less than
unity) forms a layer of solute enriched liquid. If the solute is unable to diuse away
from the interface suciently rapidly, an unstable situation may arise if the freezing
temperature of the enriched liquid rises above the actual temperature of the liquid.
Thus the liquid adjacent to the interface may become supercooled solely by a rise
in its freezing temperature due to a change in its chemical composition. Note that
instability can even occur in an isothermal situation. (In contrast, the instability
mechanism in thermal supercooling requires a reduction in the actual temperature
of the liquid below its freezing temperature.) In their pioneering work on morphological instability, Mullins and Sekerka [94] realized that the constitutional supercooling
theory neglected the eects of latent heat, capillarity and unequal thermal conductivities in the liquid and solid. Their linear stability analysis resulted in an improved
criterion which included these eects. This theory and other issues are discussed in
the comprehensive reviews of morphological stability by Coriell and McFadden [146]
and Coriell et al. [147].
Recent numerical simulations on the interface morphology in the directional solidication problem have attempted to overcome the many simplifying assumptions
66
of the previous analytical models to provide detailed quantitative predictions. Following [38] four common models used in numerical studies of directional solidication
and their associated assumptions are listed.
1. The one-sided solutal model [148], solves for the solute concentration in the
liquid only, the diusivity in the solid is assumed to be zero. It assumes a
linear temperature eld in the growth direction that is unaected by changes in
interface shape. This assumption implicitly neglects latent heat and dierences
in solid and liquid thermal conductivities.
2. The two-sided solutal model solves for the solute concentration elds in both
solid and liquid including the eect of unequal diusivities. Otherwise the
model incorporates the same linear temperature assumption as the one-sided
solutal model.
3. The two-sided thermal/one-sided solutal model solves for the temperature in
both solid and liquid including the eect of latent heat and unequal thermal
conductivities. The solute concentration is solved for only in the liquid.
4. The full thermal-solutal model solves for both solute concentration and temperature in the solid and liquid. It includes the eects of latent heat and unequal
diusivities and thermal conductivities.
Most numerical simulations of directional solidication have concentrated on obtaining steady-state solutions for the various models listed above. In a series of
papers, Brown and co-workers systematically developed and used boundary conforming nite element methods and coordinate mappings to study cellular dynamics
in steady-state directional solidication. For the one-sided solutal model, Ungar and
67
Brown [33] were the rst to compute the shape of two-dimensional solidifying cells
at steady-state and provide a systematic study of their bifurcation behavior. The
calculations in [33] were limited to relatively shallow cell shapes. In order to model
deep cells with nearly vertical or even reentrant grooves Ungar and Brown [34] used
patches of dierent interface representations for dierent portions of the interface.
For the two-sided solutal model their calculations showed the formation of dropshaped structures at the bottoms of deep cell grooves. Kessler and Levine [149]
computed similar structures using boundary integral techniques for the two-sided
solutal model including anisotropic surface tension. Experiments [102] have shown
that drops of solute enriched liquid periodically pinch-o from these deep grooves.
Tsiveriotis and Brown [35] developed a new boundary-conforming mapping method
designed to yield a co-ordinate transformation and a mesh that deforms into the
grooves of deep cellular structures. They tested this new mapping method by repeating the earlier calculations of [33] and [37]. They also developed a new nite
element method using locally rened grids to study the two-sided solutal model [36].
The full thermal-solutal model was considered by Ungar et al. [38] to study the
eects of solid diusivity and heat transfer. They found that the growth of large amplitude cells is retarded by large values of the solid diusivity and that latent heat
release at high growth rates caused local thermal undercooling at the cell tips. They
emphasized that that thermal conductivity, solid diusivity and even small amounts
of latent heat release had signicant eects on the interface structure indicating that
quantitative predictions for cellular morphologies are highly sensitive to the solidication model used. McFadden and Coriell [44] studied the eect of heat transfer
using the steady-state, two-sided thermal/one-sided solutal model for an aluminumsilver alloy. They also noted the distortion of isotherms near the interface caused by
68
latent heat release. For the one-sided solutal model McFadden et al. [45] extended
this numerical analysis to three dimensions and calculated a variety of cell shapes
including nodes and bands near the onset of instability. McCartney and Hunt [150]
used a nite dierence method with a nonuniform mesh to treat the steady growth
of an axisymmetric cell shape for the two-sided thermal/one-sided solutal model including interface kinetics. In their method, the mesh spacings are chosen such that
the interface always lies on mesh points.
Up to now only steady-state calculations have been discussed. Time-dependent
calculations have proven to be more dicult and are not as common due to the
added complexity and computational cost. Tsai and Rubinsky [39, 40] developed a
deforming nite element method to track the interface. They studied temperature
and concentration perturbations on the short time evolution of interfacial instabilities
using the two-sided thermal/one-sided solutal model. Building on previous work with
steady-state simulations, Ungar et al. [41] developed techniques for solving the timedependent two-sided solutal model. Using this time-dependent method, Bennett and
Brown [43] analyzed the complex time-periodic interactions of multiple shallow cells
for the two-sided solutal model with equal solid and liquid diusivities.
Despite the tremendous progress made in the study of cellular dynamics using the
boundary conforming techniques mentioned above, the shape and complexity of the
interface is severely restricted due to the need to construct a distorted yet acceptable
computational mesh. For time-dependent solutions this mesh must be reconstructed
at each time step. Phase-eld models [21, 23] mark a conceptual departure from
these boundary conforming methodologies. Instead of applying boundary conditions
on either side of a mathematically sharp interface having zero thickness, a scalar
phase-eld variable is used to describe the state, either liquid or solid, of various
69
regions of the system. The interface is represented by a smooth transition of the
phase-eld variable across a thin layer of nite thickness between the two phases.
The advantages of this approach are that the method can be implemented on a
simple, stationary, uniform mesh and that the interface is not tracked but given implicitly by a contour of the phase-eld variable. Thus topology changes can easily be
represented. Time-dependent methods based on phase-eld models for pure materials including anisotropy [26] have enjoyed considerable success in simulating complex
dendritic structures [18{20]. Wheeler et al. [24] and Murray et al. [25] have studied
the growth of anisotropic nickel dendrites and compared their results with sharp
interface theories for dendritic growth. Murray et al. [27] have recently simulated
the qualitative features of experimentally observed dendritic structures including the
formation of regularly spaced sidearms in tip forcing experiments. The phase-eld
method has been extended to alloy solidication by Wheeler et al. [28, 29] who derived and analyzed two phase-eld models for an isothermal system. Caginalp and
Xie [30] developed a phase-eld model for nonisothermal alloy solidication. Warren
and Boettinger [31] synthesized phase-eld models for pure material [23] and isothermal alloy solidication [28] into a formulation for nonisothermal alloy solidication.
Specializing to an isothermal system they then computed realistic features of the
growth of dendrites into a highly supersaturated liquid and studied the microsegregation patterns remaining in the solid.
The method used in this work is a straightforward extension of the numerical
method developed in chapter II for the dendritic solidication of pure materials (see
also [151]) and was motivated by front tracking techniques for multiuid ows without phase change [65, 66]. A similar technique is developed in chapter IV for the
study of phase change problems with uid ow (see also [152, 153]). Here, results
70
are presented from two-dimensional direct numerical simulations of the directional
solidication of dilute binary alloys. A direct approach to the numerical simulation
of the governing phenomenological equations and interface conditions is taken here.
Solutions for the heat and solute equations and interface motion are coupled at each
time step. Front tracking explicitly provides the location of the interface at all times,
however, the complexities associated with boundary conforming methodologies are
avoided. The method is general in the sense that complex, convoluted interfaces and
topology changes can easily be represented. Latent heat, unequal material properties between liquid and solid and anisotropic capillarity and interface kinetics are
included in the simulations. The governing equations are discretized and solved on
a simple, stationary, uniform mesh. The interface is represented by separate moving
computational points. Like in the phase-eld model, the interface is not kept sharp
but is represented on the stationary mesh by a smooth transition across a thin layer
of nite thickness. To distinguish between liquid and solid regions, a scalar indicator
function is constructed from the known position of the interface (see appendix A.4).
In contrast, the phase-eld method uses a similar variable to determine the position
of the interface.
In section 3.2 the formulation of the alloy solidication problem is presented and
in section 3.4 the method is validated through grid resolution studies and comparison
with the Mullins-Sekerka linear stability theory. Numerical results for complex microstructures demonstrate the eects of latent heat and anisotropic surface tension
on the interface morphology and solidied microstructure.
71
3.2 Mathematical Formulation
For the alloy solidication problem, the governing solute and energy equations
are written for both the liquid and solid phases simultaneously. In writing this single
eld representation the eect of the interface between the phases and the jump in
material properties from one phase to another is implicitly included. The densities of
the liquid and solid phases are assumed to be equal and constant. Volume contraction
and expansion, as well as uid convection eects are thereby neglected. These eects
are addressed in chapter IV for the problem of liquid-vapor phase change with uid
ow. The thermal conductivity, volumetric specic heat and chemical diusivity of
each phase are constant but generally not equal. The energy equation formulation,
Eqs. (2.1)-(2.4), is the same as presented in section 2.2. In addition to Eqs. (2.1)(2.4) an equation for the solute concentration must be added to the formulation.
The solute equation is usually written for both phases separately as
@Cs = r D rC and @Cl = r D rC
s
s
l
l
@t
@t
(3.1)
where C is the solute concentration and Ds and Dl are the chemical diusivities of
the solid and liquid respectively. The solute balance at the interface then requires
that
[DlrCl , Ds rCs] n = Cl(k , 1)V :
(3.2)
k = Cs (xf (t))=Cl(xf (t))
(3.3)
where
is the partition coecient and V = (dxf =dt) n is the normal velocity of the interface.
In order to recast Eqs. (3.1) and (3.2) in the form of a single equation for
both phases, a simple transformation for the solute concentration and diusivity is
72
dened [154]
8
>
>
<
Cs =k; in the solid;
C~ =
Cl; in the liquid
(3.4)
D~ s = kDs ; D~ l = Dl ;
(3.5)
>
>
:
and
then the single eld representation is
@ C~ = r D~ rC~ + S :
@t
(3.6)
S is a volumetric solute source term which accounts for rejection or absorption of
solute at the interface due to the dierence in miscibility of the alloy components in
the liquid and solid
S=
Z
A
s (x , xf ) dA
(3.7)
where s is the source of solute at the interface per unit area
s = C~f (1 , k) V :
(3.8)
C~f = C~ (xf (t)) is the value of the transformed concentration at the interface. Note
that the transformed concentration is continuous at the interface. The transformation, Eq. (3.4), is essentially the same one used by Ungar et al.. [41] for the identical
purpose of obtaining a continuous concentration throughout the computational domain. This ensures that C~f can be easily calculated numerically. (Eq. (3.5) was not
used in [41] but is included here since diusion in the solid is not neglected.) It must
be noted, however, that in using this transformation the partition coecient, k, in
Eq. (3.3) is assumed to be constant. This coecient is generally a complex function of concentration over the entire alloy phase diagram, but for the dilute alloys of
interest here the assumption of constant k is usually valid.
73
Since the interface is tracked the original concentration and diusivity elds can
be regained from the known position of the interface by
C = C~ + kC~ , C~ I (x; t)
and
(3.9)
~
(3.10)
D = D~l + Dks , D~l I (x; t) :
I (x; t) is a material indicator function similar to the phase-eld variable in phase!
eld models, that has the value 1 in the solid phase and 0 in the liquid phase. Unlike
in phase-eld methods I (x; t) is determined from the known position of the tracked
interface rather than used to determine the position of the interface. This function
also enables the evaluation of the values of the thermal conductivity and specic heat
at every location by
K (x) = Kl + (Ks , Kl) I (x; t)
(3.11)
and similarly for the specic heat. The details of the numerical construction of the
indicator function from the tracked interface are described in appendix A.4.
As a check of the consistency of this transformed single eld formulation it is easy
to show that Eqs. (3.1) and (3.2) are recovered by substitution of Eqs. (3.4) and
(3.5) into Eq. (3.6). It is important to note that Eq. (3.6) naturally incorporates the
correct solute balance at the interface. Integration of Eq. (3.6) across the interface
and substitution of Eqs. (3.4) and (3.5) yields Eq. (3.2).
In addition to the governing equations, an interface condition on the temperature
must be satised at the phase boundary [131]
Tf , Tm , mC~f + (nL)Tm + 'V(n) + Tm (cLl , cs) Tf ln TTf + Tm , Tf = 0 (3.12)
o
o
m
where Tf = T (xf (t)) is the interface temperature, (n) and '(n) are the anisotropic
surface tension and kinetic mobility coecients respectively. is twice the mean
74
curvature which is positive when the center of curvature lies in the solid phase. The
constant, m, is the slope of the liquidus line. m is usually considered to be constant
for dilute alloys but this restriction is, in general, not necessary in this formulation.
Eq. (3.12) states that the temperature at which the material changes phase depends
on: the solute concentration of the liquid at the interface, the curvature of the
interface, interface kinetics and the dierence in specic heats of the two phases.
The kinetic mobility is intended to model the nonequilibrium nature of the phase
change process and here it is assumed that kinetic eects are linearly proportional
to the interface temperature. The term containing '(n) is not included in [131].
The governing equations and boundary conditions can be made dimensionless
by scaling length by a suitable length scale, l, time by l2=Dl , velocity by Dl=l,
concentration by Co and temperature (measured from Tm) by the freezing range of the
alloy, To. From the phase diagram geometry for dilute solutions and constant m and
k, it can easily be shown that To = mCo(k , 1)=k. To limit the scope of this work
and to simplify the presentation only the case of equal specic heats is considered
here. A demonstration and discussion of the eect of discontinuous specic heat
between the liquid and solid phases on the solidication of pure materials is given in
section 2.4 (see also [151]). The chemical diusivity and thermal conductivity elds
are scaled by Dl and Kl respectively
s
D=1+ D
Dl , 1 I (x; t) ;
s
K =1+ K
Kl , 1 I (x; t) :
(3.13)
Eqs. (2.1)-(2.3), (3.6)-(3.8) and (3.12) become
@T = 1 r K rT + Q ;
@t Le
St
Q=
Z
A
q (x , xf ) dA ;
(3.14)
(3.15)
75
q=V ;
@ C~ = r D~ rC~ + S ;
@t
S = s (x , xf ) dA ;
Z
A
(3.16)
(3.17)
(3.18)
s = C~f (1 , k) V ;
(3.19)
(n) V = 0 :
n (n)
Tf + 1 ,k k C~f + kPe
+ #nPe
(3.20)
where the Lewis number is Le = clDl=Kl , the Stefan number is St = clTo=Lo,
the solutal Peclet number is Pe = Vo l=Dl, the Sekerka capillary stability parameter
is = kTmVo=Lo ToDl , and the nondimensional inverse kinetic mobility is # =
Vo ='To.
Following [19, 24, 26] the functional form of the two-dimensional anisotropic stability parameter, n(n), and inverse kinetic mobility, #n(n), is written as
n() = [1 + s cos (ns ( + s ))]
(3.21)
#n () = # [1 + k cos (nk ( + k ))] :
(3.22)
The constants s and k range between 0 and 1 and determine the magnitude of
anisotropy, ns and nk determine the mode of symmetry of the crystal and s and k
determine the angle of the symmetry axis with respect to the x,axis.
3.3 Numerical Method
The set of nondimensional equations, Eqs. (3.13)-(3.20), is solved for the ve
unknowns, T , C , V , D and K , using the two-dimensional, nite dierence/front
tracking method developed in chapter II) for the the solidication of pure materials
(see also [151]). The binary alloy problem uses a similar solution methodology with a
few straightforward modications for the addition of the solute equation, Eq. (3.17),
and its eect on the Gibbs-Thomson interface condition, Eq. (3.20).
76
3.3.1 Discretization
The energy and solute equations, Eqs. (3.14) and (3.17), for both the solid and
liquid phases are solved on a regular, xed grid using a conservative, second order,
centered dierence scheme for the spatial variables and an explicit, rst order, forward Euler time integration method. The method is traditionally called the forward
in time, centered in space, or FTCS, scheme. A variable time step is used that depends on the mesh size and the normal interface velocity, V . The maximum value of
the time step must satisfy the stability requirement for the two-dimensional FTCS
scheme. For stability, the time step, t, is required to satisfy
t h2Le=4K
(3.23)
where K is the thermal conductivity of the solid or liquid, whichever is larger and
also
t h2=4D
(3.24)
where D is the diusivity of the solid or liquid, whichever is larger. Most metal
alloys have Lewis numbers on the order of 10,4 while organic mixtures have Lewis
numbers on the order of 10,1 , 10,2 . Thus for metals numerical stiness problems
are encountered when Eqs. (3.14) and (3.17) are integrated. Since a simple explicit
time integration is used the solution is restricted by Eq. (3.23) to very small time
steps. For future calculations implementation of an implicit time integration method
would lift this restriction and allow signicantly faster computations.
The interface is explicitly tracked by using separate, non-stationary computational points connected to form a one-dimensional front which lies within the twodimensional stationary mesh. The details of the interface representation, the distribution and interpolation technique used to transfer information between the front
77
and the stationary grid and the construction of the indicator function, I (x; t) are
common to the three problems studied in this thesis and are discussed separately in
the appendix.
3.3.2 Solution Procedure
In order to begin the computation an initial interface shape is specied. From
this shape the indicator function is constructed as described in appendix A.4 and the
thermal conductivity and diusivity elds are determined from Eq. (3.13). Given
the initial interface shape, temperature, concentration and material property elds,
the solution algorithm proceeds iteratively through the following steps:
1. Using the current value of the interface velocity, the interface is advected to a
new position by V (t) = (dxf =dt) n.
2. With the estimate for the updated normal velocity, V (t +t), from Eq. (2.21),
the heat source at the interface, q, is calculated from Eq. (3.16) and is distributed to the stationary grid using Eq. (A.11).
3. C~f is found by interpolating the concentration eld found in the previous iteration onto the interface by Eq. (A.12). Using this value of C~f and the updated
normal velocity, V (t +t), the source of solute, s, is calculated from Eq. (3.19)
and is distributed to the stationary grid using Eq. (A.11).
4. With appropriate wall boundary conditions, Eqs. (3.14) and (3.17) are solved
for the temperature and concentration elds at time t + t.
5. These updated temperature and concentration elds are interpolated by Eq.
(A.12) to nd the temperature, Tf , and concentration, C~f , at each point on
the interface.
78
6. If the Gibbs-Thomson condition, Eq. (3.20), is satised then using Eq. (3.13)
the thermal conductivity and diusivity elds are updated to the interface
position found in step 1 and the computation proceeds to the next time step.
Otherwise, a new estimate for the updated normal velocity, V (t +t), is found
at each interface point using Eq. (2.21) and the procedure returns to step 2.
In the last step, the new estimate for V can be found by an iterative method. In
general, if the interface temperature and concentration found in step 5 are substituted
into Eq. (3.20) the right hand side of this equation will not equal zero but some
residual error, E(V). In order to make this error go to zero and thus satisfy Eq.
(3.20) the iteration method described in section 2.3.3 is used.
3.4 Results and Discussion
The simulations of directional solidication of a binary alloy are performed in a
two-dimensional rectangular domain which is periodic in the x-direction. The upper
portion of the domain initially contains liquid which is separated from the solid in the
lower portion by the interface. In order to simulate the temperature conditions in a
directional solidication apparatus, dimensionless temperature gradients, Gl and Gs,
are maintained at the top (y = Hy ) and bottom (y = 0) boundaries of the domain
respectively. The solute boundary condition at these boundaries is @C=@y = 0.
The initial solid-liquid interface (xf ; yf ) is nearly planar with a small amplitude
cosine perturbation described by
yf = yc + f cos (2nf xf =Hx )
(3.25)
where yc, f , nf and Hx are the average initial interface height, perturbation amplitude, perturbation mode and domain width respectively. The initial dimensionless
79
concentration eld is
C (x; y) = kC~ (x; y) = 1
(3.26)
C (x; y) = C~ (x; y) = 1 + 1 ,k k e,Pe(y,yf )
(3.27)
in the solid and
!
in the liquid. The initial temperature elds are linear in each phase
T (x; y) = Gs (y , yf ) + k ,1 1 in the solid;
T (x; y) = Gl (y , yf ) + k ,1 1 in the liquid :
(3.28)
(3.29)
These initial conditions are simply the solution to Eqs. (3.14)-(3.20) for a steadily advancing planar interface at velocity Pe neglecting kinetic eects, # = 0 and capillary
eects, = 0.
Mullins and Sekerka [94] performed a linear stability analysis of the growth of an
innitesimal sinusoidal perturbation of amplitude on a steadily advancing, otherwise planar interface. Assuming that there is no diusion in the solid, Ds = 0, they
obtained the following dispersion relation between the growth rate of the perturbation amplitude, , and the wave number, ! = 2=
1 d = 1 , 2k=
, 1=S , 2=k
= Pe
dt Le= (
St (Ks =Kl + 1)) + 2=
(3.30)
where = !=Pe and = (1 + 4
2)1=2 + 2k , 1. If the numerator of the right side
of Eq. (3.30) is positive for any wave number then the interface is unstable. The
criterion for instability is that
SF (; k) > 1
(3.31)
where the Sekerka number [155] is S = Pe (Ks =Kl + 1) = (Gs Ks =Kl + Gl) and the
stability function F is plotted in [156] for various values of and k. An interesting
80
limit is when the capillary length is of the same magnitude as the solute diusion
length (at high interface velocities [157], for example). Then approaches unity, F
approaches zero and the interface will be \absolutely stable".
In contrast to the Mullins-Sekerka criterion, the constitutional supercooling criterion is clearly less restrictive since it neglects latent heat (G = Gs = Gl), capillarity
(F = 1) and assumes that the thermal conductivities are equal (Ks = Kl ). Instability
is predicted for
Pe > 1 :
G
The simulation in Fig. (3.1) was run with the following parameters
(3.32)
Dl = 1000; Kl = 1; cl = 1; k = 0:4
Ds
Ks
cs
Pe = 1; Le = 0:001; St = 0:01; = 0:004 # = 0:002
Gl = 0:1 Gs = 0:2 :
The computational domain is 2 where is the wavelength of the fastest growing
unstable mode of the Mullins-Sekerka instability. For this case = 1:1766. The
initial interface shape and location is given by Eq. (3.25) with yc = 0:3, f =
,0:01 and nf = 1. As the liquid solidies the interface exhibits the growth of the
Mullins-Sekerka instability. As a test of grid renement, results from three dierent
resolutions are superimposed at t = 1:2. The coarse resolution, 10 20 is clearly
underresolved. The results on the two ner grids are nearly identical indicating that,
for this choice of parameters, the 20 40 and 40 80 grids provide a converged
solution.
Mullins and Sekerka neglected solid diusion, interface kinetics and unsteady effects and assumed that the interface moves with a constant mean velocity. None of
81
these assumptions are made in the present numerical model. However for comparison, parameters are chosen to approximately match the assumptions made in the
linear theory. The best agreement would be expected early in the calculation before
nonlinear eects come into play. In Fig. (3.2) the interface perturbation amplitude,
, predicted from linear theory is compared with the numerical results for dierent
grid resolutions from Fig. (3.1). The results match closely at early times (small amplitudes) but deviate at later times as the interface amplitude grows. The numerical
results also more closely match the linear theory for higher resolution.
Fig. (3.3) plots the dierence in the amplitude, , between the numerical results
and linear theory at t = 0:45 for several dierent grid resolutions. The numerical
results exhibit linear convergence at higher resolutions. Consistent with the study
by LeVeque and Li [136] linear convergence is observed in the dendritic solidication
results in Chapter II and here the results for alloy solidication also exhibit linear
convergence.
The simulation in Fig. (3.4) was run in a 2 10 domain with a grid resolution
of 50 200. The interface starts with two cells at yc = 0:3 with f = ,0:01 and
nf = 2. The other parameters are the same as in Fig. (3.1) except the Peclet number
is increased to, Pe = 10. In order to promote preferred growth in the horizontal and
vertical directions anisotropic capillarity is included by s = 0:4, ns = 4 and s = 45o.
Fig. (3.4) shows ten frames from the computation at dierent times. The white line
is the solid-liquid interface and the lighter shades of gray represent higher solute
concentration. The solute concentration color scale for this gure is the same as the
one shown in Fig. (3.6). The initial concentration in the solid is given by Eq. (3.26).
82
However, for this simulation the concentration prole in the liquid is
C (x; y) = C~ (x; y) = 1 + 1 ,k k e,(y,yf )
!
(3.33)
which is slightly dierent than the fully developed prole given by Eq. (3.27). The
initial concentration gradient in the liquid at the interface as specied by Eq. (3.33)
is ten times less than the fully developed prole and a transient period of solute
build up ahead of the advancing interface is expected. As solidication progresses,
this solute build up during the initial transient leads to onset of the Mullins-Sekerka
instability. Frame (c) shows rapid growth of granular protrusions into the liquid. This
necking phenomenon is often seen near the mold wall of castings [158]. In the narrow,
necked region of the interface the higher solute concentrations promote melting and
eventual pinch-o of the granules. In actual castings this narrow necked granular
crystal is also easily separated by uid forces due to vibrations, natural or forced
convection. Depending on the density of the granule relative to the surrounding melt,
this particle is generally transported to other parts of the mold to act as a nucleation
site or to precipitate further separation of similar granules. This mechanism of crystal
multiplication greatly eects the nal solid microstructure.
In this simulation there is no uid ow so the separated granules remain above a
trapped pocket of high concentration liquid and continue to grow upward more slowly
in a classic, regular, cellular pattern. The cells exhibit a typical cored structure [158]
with low concentration solid at the cell's central axis and progressively higher concentration laterally toward the cell surface. High concentration liquid drops periodically
pinch-o from the deep intercellular grooves and subsequently solidify leaving behind
a trail of vertical bands of high concentration solid. The cell tips contain solid of low
solute concentration and leave behind a vertical band of low concentration solid.
83
Between frames (f) and (g) the temperature gradient at the bottom boundary is
increased from Gs = 0:2 to 2. This has the eect of accelerating the solidication
process. The previously trapped liquid is now able to solidify at higher concentration
and the intercellular grooves recede upward to form a at stable interface. In the
last frame, (j), the entire domain is nearly completely solid. The solute inhomogeneities, or microsegregation patterns, frozen into the solid are what characterize
the microstructure and physical properties of the solid. Note the microsegregation,
evident as bands of higher concentration solid: the broad horizontal band which has
solidied from the trapped high concentration liquid pocket (a result of necking),
and the vertical bands of solid parallel to the direction of cell growth, (a result of
coring).
An important contribution of this method to the study of alloy solidication is
the ability to solve for both the temperature as well as the solute concentration. Fig.
(3.5) corresponds to frame (c) of Fig. (3.4) and illustrates the eect of latent heat
release at the interface on the heat ow during rapid interface growth at t = 0:1.
The common assumption of an imposed linear temperature gradient would lead to
dramatically dierent results. The direction of heat ow indicates that the cell tips
are growing into an undercooled liquid, a highly unstable situation. Simulations by
Ungar et al. [38] and McFadden and Coriell [44] which included the eect of heat
transfer but for steady-state growth also demonstrated that latent heat release at
high growth rates caused local thermal undercooling at the cell tips.
The eect of capillary anisotropy is demonstrated in Fig. (3.6). The left hand
frame is a calculation at time, t = 0:47, performed with the same initial conditions
as in Fig. (3.4). The right hand frame is a repeat of the calculation with s = 0o
and is shown for the same time. The preferred growth direction in this frame is
84
at a 45o angle to the x,axis and the eect is to produce a more highly distorted
liquid-solid interface with no regular cellular growth and a qualitative change in the
solid microstructure.
The formation of alloy dendrites during directional solidication is shown in Fig.
(3.7). The detailed dendritic structure is resolved with a 200 600 grid in a 0:5 1:5
domain. At this resolution diculties are encountered due to the severe time step
restriction imposed by Eq. (3.23) for realistic values of the Lewis number. This numerical stiness problem is avoided by making the assumption that the temperature
gradient, G, is linear throughout the domain. Essentially, the unsteady, two-sided
solutal model described in section 3.1 is used for this simulation. Larger time steps
can now be taken since the maximum time step need only satisfy Eq. (3.24). The
solution procedure is the same except that now the temperature eld at each time is
described by the linear prole
T (x; y) = G(y , yc , t Pe) + k ,1 1
(3.34)
instead of by Eqs. (3.14)-(3.16).
The simulation in Fig. (3.7) was run with the following parameters
Dl = 1000; k = 0:1; Pe = 1;
Ds
= 10,6 ; # = 2(10),4 ; G = Gl = Gs = 0:0146 :
In order to promote preferred growth in the horizontal and vertical directions anisotropic
capillarity is included by s = 0:4, ns = 4 and s = 45o . The initial interface is given
by Eq. (3.25) with yc = 0:05, f = ,0:001 and nf = 3. The initial concentration
eld is given by Eqs. (3.26) and (3.27). The three small perturbations on the initial
interface quickly bifurcate to form three pairs of symmetrically growing arms. In
85
order to break the symmetry and ensure that the right arms of each pair outgrow
the left, the interface is given an additional small perturbation
yf = ,0:003 cos (12xf =Hx)
(3.35)
at t = 0:08. After this time the right arms of each pair grow rapidly and undergo
repeated branching while the left arms grow much more slowly. With such a small
value of the partition coecient, k = 0:1, a large amount of solute is rejected at the
rapidly growing dendrite tips leaving behind a high concentration liquid wake which
severely slows the growth of any branches left behind. The interfaces are plotted at
equal time increments of 0.01 and the last shape at t = 0:17 contains 1,531 interface
points. Only the bottom half of the computational domain is shown in Fig. (3.7).
3.5 Conclusions
A method for time-dependent, two-dimensional solidication of dilute binary alloys is developed. This appears to be the rst method which solves the coupled,
time-dependent solute and energy equations including the eects of large interface
deformations, topology change, latent heat, anisotropic capillarity and interface kinetics and discontinuities in material properties between the liquid and solid phases.
The major contribution is that latent heat is not neglected and a known temperature
or an isothermal system is not assumed. Here the method is used to perform twodimensional simulations of directional solidication of a binary alloy. The method is
validated through comparison with the Mullins-Sekerka linear stability theory and
by grid resolution studies. Good agreement is found with the linear theory for small
interface deformations before nonlinear eects come into play. For larger growth
rates, experimentally observed complex morphological structures and solute segregation patterns are seen. The eect of latent heat release and anisotropic capillarity
86
on the interface instability and solidication microstructure is demonstrated. For
a high resolution calculation with a linear temperature gradient dendritic growth
during directional solidication is observed.
87
2
Y
1.5
1
10 x 20
0.5
20 x 40
40 x 80
0
0
0.5
1
X
Figure 3.1: A grid resolution study for alloy solidication. As the liquid solidies the
interface exhibits the growth of the Mullins-Sekerka instability. Results
from three dierent resolutions are superimposed at t = 1:2. The coarse
resolution, 10 20 is clearly underresolved. The results on the two ner
grids are nearly identical indicating that, for this choice of parameters,
the 20 40 and 40 80 grids provide a converged solution. Dl =Ds =
1000; Kl=Ks = 1; cl=cs = 1; k = 0:4; Le = 0:001; St = 0:01; =
0:004 # = 0:002; Gl = 0:1 Gs = 0:2 .
88
0.05
0.045
0.04
Linear Theory
10 x 20
0.035
20 x 40
40 x 80
Amplitude
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.05
0.1
0.15
0.2
0.25
Time
0.3
0.35
0.4
0.45
0.5
Figure 3.2: The interface perturbation amplitude, , predicted from linear theory is
compared with the numerical results for dierent grid resolutions from
Fig. (3.1). The results match closely at early times (small amplitudes)
but deviate at later times as the interface amplitude grows. The numerical results also more closely match the linear theory for higher resolution.
89
−2
10
e
nc
ic C
ge
er
drat
nv
Co
Qua
ar
ne
Li
log(error)
Numerical Results
nce
erge
onv
1
2
10
10
log(1/h)
Figure 3.3: A plot of the dierence in interface amplitude, , between the numerical
results and linear theory at t = 0:45. The numerical results exhibit linear
convergence at higher resolutions as expected with the front tracking
method.
90
Figure 3.4: Computation of complex microstructure. The white line is the solidliquid interface and the lighter shades of gray represent higher solute
concentration. Ten dierent times during the computation are shown.
Between frames (f) and (g) the temperature gradient at the bottom
boundary is increased from Gs = 0:2 to 2. In the last frame the entire domain is nearly completely solid and shows the resulting microsegregation
patterns formed during solidication. The domain is 2 10, the resolution is 50 200. The parameters are the same as in Fig. (3.1) except
Pe = 1, yc = 0:3, f = ,0:01 and nf = 2 with anisotropic capillarity,
s = 0:4, ns = 4, s = 45o .
91
4.5
4
3.5
Y
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
X
Figure 3.5: The eect of latent heat release at the interface on the heat ow at
t = 0:1. This gure corresponds to frame (c) of Fig. (3.4). The disturbance on the temperature eld caused by the rapid growth of the
interface is illustrated . The direction of heat ow indicates that the
cell tips are growing into an undercooled liquid, a highly unstable situation. An important contribution of this method to the study of alloy
solidication is the ability to solve for both the temperature as well as
the solute concentration. The common assumption of an imposed linear
temperature gradient would lead to dramatically dierent results.
92
Figure 3.6: The eect of capillary anisotropy. The left hand frame is a calculation
at time, t = 0:47, performed with the same initial conditions as in Fig.
(3.4). The right hand frame is a repeat of the calculation with s = 0o
and is shown for the same time. The preferred growth direction in this
frame is at a 45o angle to the x,axis and the eect is to produce a more
highly distorted liquid-solid interface with no regular cellular growth and
a qualitative change in the solid microstructure.
93
0.7
0.6
0.5
Y
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
X
Figure 3.7: Dendritic growth in directional solidication with a linear temperature
gradient. For this simulation it is assumed that the temperature gradient,
G, is linear throughout the domain. Essentially, the unsteady, two-sided
solutal model described in section 3.1 is used here in order to allow a
larger time step and a faster computation time. With such a small value
of the partition coecient, k = 0:1, a large amount of solute is rejected
at the rapidly growing dendrite tips leaving behind a high concentration
liquid wake which severely slows the growth of any branches left behind.
The interfaces are plotted at equal time increments of 0.01 and the last
shape at t = 0:17 contains 1,531 interface points. The grid resolution
is 200 600 grid with a 0:5 1:5 domain. Only the bottom half of
the computational domain is shown. The simulation was run with the
following parameters Dl =Ds = 1000; k = 0:1; Pe = 1; = 10,6 ; # =
2(10),4 ; G = Gl = Gs = 0:0146; s = 0:4; ns = 4; s = 45o yc =
0:05; f = ,0:001; nf = 3. The three small perturbations on the initial
interface quickly bifurcate to form three pairs of symmetrically growing
arms. In order to break the symmetry and ensure that the right arm
of each pair outgrows the left, the interface is given an additional small
perturbation, yf = ,0:003 cos (12xf =Hx) at t = 0:08.
CHAPTER IV
LIQUID-VAPOR PHASE CHANGE WITH
FLUID FLOW
4.1 Introduction
Fluid ow combined with phase change is an important part of the power generation process. The high heat transfer rates typical of boiling are used to extract
energy from solar, fossil and nuclear fuels. Designers of energy generation systems
for spacecraft must deal with the added complication of handling low boiling point
cryogenic uids in the absence of gravity. The main feature which categorizes ows
with phase change is the behavior of the interface separating the phases. For example, boiling of a liquid from a solid heated surface can be described by three regimes:
nucleate, transition and lm boiling. The interface geometry and dynamics determine to a large extent the heat transfer rates in these three boiling modes. Nucleate
boiling is characterized by individual vapor bubble formation at distinct sites on the
heated surface. Heat transfer is mainly through contact between the heated surface
and the liquid. High heat uxes are possible at relatively low surface temperatures
due to rapid mixing associated with bubble formation. In transition boiling, bubble
formation is rapid enough that a vapor blanket begins to cover the heated surface.
In lm boiling, a layer of vapor completely blankets the heated surface. A balance
94
95
is maintained between vapor generation due to vaporization at the liquid-vapor interface and vapor removal due to the break o and rise of vapor bubbles from the
interface. Normally, it is desirable to operate close to the peak nucleate boiling heat
ux. However, in processes where the heat ux is the controllable variable such
as in nuclear reactor operation or in electrically heated applications, exceeding this
critical heat ux can be dangerous. The process jumps to the lm boiling regime
where heater damage can occur due to the high surface temperatures. The vapor
layer acts as an insulator thereby lowering the heat transfer rate and increasing the
heater surface temperature.
Accidental vapor explosions due to sudden depressurization or overheating of
uid lled tanks are a serious safety concern. In vapor explosions, extremely rapid
evaporation of a liquid at the superheat limit due to sudden depressurization or contact with a hotter surface can lead to destructive accidents [1]. Several experimental
studies involving the superheating of liquid drops in bubble columns have been undertaken to characterize this phenomena [159{161]. Ervin et al. [2] have observed an
interfacial instability on the surface of bubbles in microgravity boiling experiments
on the U.S. Space Shuttle. They were the rst to report observations of this type
of interfacial instability in a bulk liquid heated from a solid surface. They believed
that the small scale protuberances on the growing bubble's surface greatly increased
the liquid-vapor interface surface area which then resulted in rapid evaporation and
the creation of more protuberances.
Although considerable experimental knowledge has also been accumulated, the
small spatial scales and the rapidity of the phase change process make it very dicult
to obtain the necessary measurements. Thus these observations tend to be mostly
visual, with measurements conned to global quantities. Furthermore, opportuni-
96
ties for experimental investigation of boiling processes under microgravity conditions
are limited to short duration experiments on Earth or to expensive space ights.
Numerical simulations hold the promise to complement experimental investigations
and provide information that is hard to measure. By understanding this small scale
information it is hoped that progress can be made toward the long-term goal of providing quantitative predictions for linking operating conditions to large scale aspects
of heat exchanger design and eciency.
Analytical and numerical eorts to understand the processes involved in boiling
have focused mainly on simple numerical and analytical models of vapor bubble
dynamics. Due to the complexity of the liquid-vapor phase change problem, an
assumed interface shape along with various assumptions concerning surface tension,
uid viscosity and vapor phase velocity and temperature are usually incorporated.
Lord Rayleigh [78] formulated a simplied equation of motion for inertia controlled
growth of a spherical vapor bubble. Rayleigh's analysis was extended by, among
others, Plesset and Zwick [79, 80], Mikic et al. [81], Dalle Donne and Ferranti [82]
and Lee [83] to include thermal and surface tension dominated growth regimes. In
recent numerical work, Lee and Nydahl [84] compute hemispherical bubble growth in
nucleate boiling from inception through departure. Patil and Prusa [85] numerically
study the thermal diusion controlled growth of a hemispherical bubble as well. The
numerical solution of the phase change problem with uid ow is particularly dicult
due to the coupling of the mass, momentum and energy transport with the interface
dynamics and since interphase mass transfer results in discontinuous velocities at
the phase boundaries. Welch [87] has made signicant progress in using a twodimensional, moving mesh, nite volume method to solve the mass, momentum and
energy equations for liquid-vapor ows with phase change. However, his method is
97
restricted to ows with only small distortion of the liquid-vapor interface. Schunk
and Rao [88] solve a multicomponent phase change problem using a boundary-tted
nite element method developed by Christodoulou and Scriven [89] for free surfaces.
The method is applied to formation of sol-gel thin lms and bres. In this case the
physics of the problem is relatively complex, however, the interface is limited to small
distortions from its initial shape.
In this chapter a numerical technique is developed for uid ow with phase change
that enables the simulation of problems with relatively complex motion of the boundary separating two uids. The method is based on a nite dierence approximation of
the Navier-Stokes and energy equations and an explicit tracking of the phase boundary. It is an extension of techniques already developed for multiuid ows without
phase change by Unverdi and Tryggvason [65,66]. The multiuid code has been used
to investigate the collision of drops [74, 134], thermal migration of drops [135] and
the motion of several bubbles [162].
In the next two sections the mathematical formulation and numerical method
for the liquid-vapor phase change problem is presented. The eects of interphase
mass transfer, latent heat, surface tension and unequal material properties between
liquid and vapor phases are included. Results from two-dimensional simulations of
lm boiling from an upward facing at surface in a horizontally periodic domain
demonstrate the ability of the front tracking method to easily handle large interface
deformations and topology change. The liquid-vapor interface exhibits a RayleighTaylor instability, with subsequent pinch-o and rise of a vapor bubble. The hot
vapor from regions near the wall is convected up into the bubble and is carried
upward into the ambient uid with the bubble. A balance is maintained between
vapor generation due to vaporization at the liquid-vapor interface and vapor removal
98
due to the break o and rise of vapor bubbles from the interface. Heat transfer
results show good agreement with a lm boiling correlation by Klimenko [163] for
several dierent uids.
The rapid evaporation problem is also studied and the energetic growth of instabilities on planar and circular interfaces during the unstable, explosive evaporation
of a superheated liquid in microgravity is demonstrated. Simulations show the formation of highly convoluted interfaces and enhancement of evaporation leading to
explosive growth.
4.2 Mathematical Formulation
The liquid-vapor phase change problem involves uid ow as well as heat transfer.
This requires the solution of the Navier-Stokes and energy equations. However mass
transfer across the interface and momentum as well as energy sources at the interface
must now be taken into account. Note that in two-phase ow, additional terms
appear in these equations due to the phase change and the fact that the interface is
no longer a material interface. The uid velocity at the interface and the interface
velocity are unequal.
As before a single set of governing equations is written for both phases. This
local, single eld formulation incorporates the eect of the interface on the governing
equations as sources which act only at the interface. Kataoka [164] shows that this
local, single eld representation is equivalent to the local, separate phase formulations
of Ishii [165] and Delhaye [166]. They formulate the phase change problem in terms
of variables for each phase with appropriate jump conditions at the moving phase
interface. Those local, separate phase formulations form a fundamental basis for all
averaged models of two-phase mixtures.
99
Here the single eld, local formulation is used. The material properties are considered to be constant but not generally equal for each phase. They can be written
for the entire domain and advected using the indicator function, I (x; t). This function is constructed from the known position of the interface and has the value 1 in the
vapor phase and 0 in the liquid phase. The numerical construction of the indicator
function is discussed in appendix A.4. The values of the density, viscosity, specic
heat, and thermal conductivity elds at every location are then given by
(x) = l + (v , l) I (x; t) ;
(4.1)
(x) = l + (v , l) I (x; t) ;
(4.2)
c(x) = cl + (cv , cl) I (x; t) ;
(4.3)
K (x) = Kl + (Kv , Kl) I (x; t) ;
(4.4)
respectively, where the subscripts v and l refer here to the vapor and liquid phases
respectively.
The momentum equation is written for the entire ow eld and the forces due
to surface tension are inserted at the interface as body forces which act only at the
interface. In conservative form this equation is
@ w + r (wu) = ,rP + g + r ru + ruT + F :
@t
(4.5)
Here u is the uid velocity, w = u is the mass ux, P is the pressure and g is the
gravity vector. F is a volumetric source term which accounts for forces acting on the
interface
Z
F = A f (x , xf ) dA :
(4.6)
100
(x , xf ) is a three-dimensional delta function that is non-zero only at the interface
where x = xf . f is the force per unit area normal to the interface (surface tension),
f = n
(4.7)
where is the surface tension coecient and is twice the mean curvature. Thermocapillary forces acting in a direction, t, tangential to the interface could arise from
variation of the surface tension with temperature, but this eect is neglected here.
The conservation of mass equation is also written for the entire ow eld
@ + r w = 0 :
@t
(4.8)
The time derivative of the density can be rewritten in a more useful form since its
value at each point in the domain, Eq. (4.1), depends only on the indicator function
which is determined by the known interface location. Using the indicator function,
I (x; t), to represent the interface, the kinematic equation for a surface moving with
velocity, V, is
@I = ,V rI ;
(4.9)
@t
Using Eqs. (4.9), (A.16) and Eq. (4.1) for the density, the conservation of mass, Eq.
(4.8), can be rewritten as
rw =M
(4.10)
where M is the volumetric mass exchange at the interface due to the phase change
(volume expansion at the interface)
M=
Z
A
m (x , xf ) dA
(4.11)
and m is the mass transfer across the interface per unit area due to the phase change
m = (v , l) V n :
(4.12)
101
The thermal energy equation is [167]
@ (cT ) + r (wcT ) = r K rT , T @P
@t
@T
Dv^ + T Dc
Dt
v^ Dt
!
(4.13)
where T is the temperature and
v^ = v^l + (^vv , v^l)I (x; t)
(4.14)
is the specic volume. The viscous dissipation has been neglected. Using Eqs. (4.3),
(4.9) and (4.14) the material derivatives, Dc=Dt and Dv^=Dt, can be written as
Dc = (c , c ) (u , V) rI
v
l
Dt
(4.15)
and
Dv^ = (^v , v^ ) (u , V) rI :
(4.16)
v
l
Dt
In addition, the Clausius-Clapeyron relation along the pressure-temperature saturation curve gives
@P = dP = L + (cv , cl) Tf
(4.17)
@T v^ dT sat
(^vv , v^l) Tf
where Tf = T (xf (t)) is the interface temperature. In this equation the latent heat,
!
!
L, takes into account unequal specic heats
L = Lo + (cl , cv ) Tv ;
(4.18)
where Lo is the customary latent heat measured at the reference equilibrium vaporization temperature, Tv .
Using Eqs. (4.15)-(4.17), the thermal energy equation, Eq. (4.13), can be rewritten as
@ (cT ) + r (wcT ) = r K rT + Q
@t
(4.19)
102
where Q is a volumetric energy source term which accounts for the liberation or
absorption of latent heat, L, at the liquid-vapor interface
Q=
Z
q (x , xf ) dA :
A
(4.20)
q is the energy source at the interface per unit area
q = L (V , w) n :
(4.21)
It is important to recognize that the single eld formulation (Eqs. (4.5), (4.10)
and (4.19)) naturally incorporates the correct mass, momentum and energy balances
across the interface since integration of these equations across the interface directly
yields the jump conditions derived in the local, separate phase formulation for twophase systems given by Delhaye [166] and Ishii [165]. These are the mass jump
condition
[ w] n , [ ] V n = 0 ;
(4.22)
the momentum jump condition normal to the interface
[ wu] n , [ w] V n + P , ru + ruT
hh
ii
n = n ;
(4.23)
the momentum jump condition tangential to the interface
ru + ruT t = 0 ;
hh
ii
(4.24)
and the thermal energy jump condition
[ K rT ] n = [Lo + (cv , cl) (Tf , Tv )] (wv , v V) n
(4.25)
where [ ] denotes the jump in a quantity from the vapor to the liquid side of the
interface. For Eq. (4.24) a no-slip condition is assumed for the tangential uid
velocities across the interface, [ u] t = 0, and no tangential variation in the surface
103
tension (no thermocapillary eects). In addition, it is assumed that the interface is
thin and massless and that the bulk uids are incompressible. In the energy equation,
viscous dissipation and kinetic energy contributions from the product of the uid
velocity at the interface and the interface velocity are neglected. Contributions to
the source term in the energy equation from interface stretching are usually small
compared with the latent heat and are neglected.
To complete the formulation a condition on the temperature at the phase change
interface must be specied. In recent studies on interface instability during phase
change, Huang and Joseph [168, 169] point out that the correct condition for the
temperature at a phase change boundary is not known and is still an unresolved
physical issue. They note that thermodynamic equilibrium (the Clausius-Clapeyron
relation, Eq. (4.17)) excludes thermal equilibrium (continuity of temperature at the
interface, [ T ] = 0). Typically it is assumed that the vapor and liquid temperatures
at the interface are equal and the value of this interface temperature determined by
the Clausius-Clapeyron relation for the saturation value appropriate to the pressure
in the vapor. But since the pressures on either side of the interface are generally
not equal (see Eq. (4.23)), the liquid temperature at the interface is not given by
the Clausius-Clapeyron relation. Thus the liquid at the interface is not in thermodynamic equilibrium.
For the numerical calculations in this work, thermal equilibrium at the interface,
[ T ] = 0, is assumed, but not thermodynamic equilibrium. The value of the interface temperature, Tf , is found using a slight variation of the interface temperature
condition derived by Alexiades [131] from a careful consideration of the equilibrium
104
Clausius-Clapeyron relation for a curved interface
(4.26)
Tf , Tv , TLv + LTv 1 , 1 (Pv , P1 )
l o
o
l
v
, (cl , cv ) LTv Tf ln TTf + Tv , Tf + (V ,'u) n = 0 ;
(4.27)
o
v
where P1 and Pv are the ambient pressure and the pressure at the interface in the
!
vapor respectively. Analogous to the kinetic mobility in solidication, the last term
on the left side of this equation is intended to model the inherent nonequilibrium
nature of the phase change process through a molecular kinetic parameter, '. It
represents a slight deviation from the equilibrium Clausius-Clapeyron relation. (This
nonequilibrium term is not included in [131].) Here the molecular kinetic eects are
assumed to be linearly proportional to the interface temperature.
The governing equations and boundary conditions can be made dimensionless by
scaling length by a suitable length scale, l, velocity by a velocity scale, Uo , gravity
by G, pressure (measured from P1 ) by lUo2 and temperature (measured from Tv )
by v Lo=l cl. The density, viscosity, specic heat and thermal conductivity elds are
scaled by l, l, cl and Kl respectively
= 1 + v , 1 I (x; t)
l
!
(4.28)
= 1 + v , 1 I (x; t)
l
c = 1 + ccv , 1 I (x; t)
l
v
K =1+ K
K , 1 I (x; t) :
!
(4.29)
(4.30)
(4.31)
l
Eqs. (4.5)-(4.7), (4.10)-(4.12), (4.19)-(4.21) and (4.27) become
@ w + r (wu) = ,rP + g + 1 r ru + ruT + F ;
@t
Fr Re
(4.32)
105
Z
F = A f (x , xf ) dA ;
f = n ;
We
rw =M ;
(4.33)
(4.34)
(4.35)
Z
m (x , xf ) dA ;
(4.36)
m = v , 1 V n ;
(4.37)
l
@ (cT ) + r (wcT ) = 1 r K rT + Q
(4.38)
@t
Pe
Q = q (x , xf ) dA :
(4.39)
A
q = lLL (V , w) n ;
(4.40)
v o
Tf , + We 1 , l Pv + v ccv , 1 Tf2 + # (V , w) n = 0 : (4.41)
v
l
l
where the Reynolds number is Re = lUol=l , the Froude number is Fr = Uo2=Gl, the
M=
A
!
Z
!
Weber number is We = lUo2l= , the Peclet number is Pe = lclUol=Kl, the capillary
parameter is = clTv =v L2ol and the dimensionless nonequilibrium parameter is
# = l clUo=v Lo'.
4.3 Numerical Method
An iterative strategy is used to solve the system of Navier-Stokes and energy
equations, Eqs. (4.32), (4.35) and (4.38). The solution to Eqs. (4.32), (4.35) is
similar to Chorin's [170] projection algorithm for variable-density ows but now
with modications to account for energy transport and phase change.
4.3.1 Discretization
Equations (4.32) and (4.35) can be simply rewritten as
@ w = A , rP
@t
(4.42)
106
rw =M
(4.43)
where the advection, diusion, gravity and surface tension terms in Eq. (4.32) have
been lumped into A. Using a rst order, explicit, forward in time method the discrete
form of the equations can be written
wn+1 , wn = An , r P ;
(4.44)
rh w = Mh :
(4.45)
h
t
h
The subscript h denotes a nite dierence approximation to the operator and n
denotes the time level. Eq. (4.45) must hold at all times. The numerical treatment
of the viscous terms in Ah is slightly unconventional and is discussed below in deriving
Eq. (4.51) for Ah.
Taking the divergence of Eq. (4.44) and using Eq. (4.45) leads to a Poisson
equation for the pressure
h + r An
r2hP = Mh ,M
h h
t
n
n+1
(4.46)
This equation is solved for the pressure using a fast Poisson solver. The updated
mass ux is then calculated by
wn+1 = wn + t (Anh , rhP ) :
(4.47)
The updated velocity is simply
un+1 = wn+1 :
n+1
(4.48)
Once the velocity is known, the discretized energy equation, Eq. (4.38), is then
solved for the temperature eld
n cnT n + t Bhn + Qnh+1
n
+1
T =
n+1cn+1
(4.49)
107
where the advection and diusion terms in Eq. (4.38) have been lumped into B
1 r K r T
Bh = ,rh (wcT ) + Pe
h
h
(4.50)
For the spatial discretization, the MAC method of Harlow and Welch [46] is
used with a staggered mesh. The pressure, temperature, and indicator function are
located at the cell centers, the x,component of velocity at vertical cell faces and the
y,component of velocity at horizontal cell faces.
The numerical treatment of the viscous term in Ah requires special consideration.
Although the uid in each phase is considered to be incompressible, a local volume
expansion or contraction at the interface is necessary during phase change to satisfy
conservation of mass, Eq. (4.35). This expansion or contraction results in a discontinuity in normal velocity at the interface. As long as the interface is considered to be
mathematically innitely thin, the discontinuity is sharp and the velocities in each
phase remain divergence free. However, in the numerical treatment the interface has
a nite thickness on the order of a few mesh blocks. The discontinuity in normal
velocity is numerically no longer sharp and is spread over several grid points near
the interface. Thus the divergence of the velocity will be locally nonzero near the
interface. This nonzero divergence contributes to the normal stress at the interface
and produces an articial, local spike in the pressure near the interface. This numerical artifact can easily be removed simply by removing the divergence terms from
the stress tensor. Thus Ah is rewritten as
g + 1 r ru + ruT , 2 (r u) l + F
Ah = ,r (wu) + Fr
Re
h
i
(4.51)
where l is the identity tensor. Subtracting twice the divergence from the stress tensor
has the eect of forcing the stress tensor to be divergence free everywhere. Also
note that if viscous dissipation had been included in the energy equation a similar
108
correction would have been necessary for that term. It is important to recognize
that the discontinuity of normal velocity at the interface during phase change is a
physically dierent phenomenon from the steep but continuous change in velocity
across a shock wave in compressible ow. The large velocity gradients across a shock
result in physically meaningful contributions to the viscous diusion of momentum
and energy.
The interface is explicitly tracked by using separate, non-stationary computational points connected to form a one-dimensional front which lies within the twodimensional stationary mesh. The details of the interface representation, the distribution and interpolation technique used to transfer information between the front
and the stationary grid and the construction of the indicator function, I (x; t), are
common to the three problems studied in this thesis and are discussed separately in
the appendix.
4.3.2 Solution Procedure
In order to begin the computation an initial interface shape is specied. From
this shape the indicator function is constructed as described in appendix A.4 and the
material property elds are determined from Eqs. (4.28)-(4.31). Given the initial
interface shape, temperature, concentration and material property elds, the solution
algorithm proceeds iteratively through the following steps:
1. Using the current value of the interface velocity, the interface is advected to a
new position by V n = (dxf =dt) n.
2. The density, n+1 , and specic heat, cn+1 are calculated for this new interface
position using Eqs. (4.28) and (4.30).
109
3. The surface tension, f , is calculated using Eq. (4.34) and is distributed to the
stationary grid using Eq. (A.11).
4. With the estimate for the updated normal velocity, V n+1, from Eq. (2.21),
the mass transfer across the interface, m, is calculated using Eq. (4.37) and is
distributed to the stationary grid using Eq. (A.11).
5. With appropriate wall boundary conditions, Eqs. (4.32) and (4.35) are solved
for the mass ux, velocity and pressure at time n + 1.
6. The density found in step 2 and the mass ux found in step 5 are interpolated
by Eq. (A.12) to nd the mass ux, wf , and density, f , at the interface.
7. The heat source at the interface, q, is then calculated using Eq. (4.40) and
distributed to the stationary grid using Eq. (A.11).
8. With appropriate wall boundary conditions, Eq. (4.38) is solved for the temperature at time n + 1.
9. The temperature and pressure are interpolated by Eq. (A.12) to nd the
temperature, Tf , and pressure, Pf , at each point on the interface.
10. If the Gibbs-Thomson condition, Eq. (4.41), is satised then the viscosity and
thermal conductivity elds are updated to the new interface position found in
step 1 by Eqs. (4.29) and (4.31) and the computation proceeds to the next
time step. Otherwise, a new estimate for the updated normal velocity, V n+1,
is found at each interface point using Eq. (2.21) and the procedure returns to
step 4.
In the last step, the new estimate for V can be found by an iterative method. In
general, if the interface temperature and pressure found in step 9 is substituted into
110
Eq. (4.41) the right hand side of this equation will not equal zero but some residual
error, E(V). In order to make this error go to zero and thus satisfy Eq. (4.41) the
iteration method described in section 2.3.3 is used.
Note that the interpolated pressure, Pf , found in step 9 is not the pressure on
the vapor side of the interface required in Eq. (4.41). If the pressure jump across
the interface is not too large then this should not be a signicant discrepancy. In
general the value of a discontinuous interfacial quantity on one side or the other of
the interface is dicult to determine numerically. This issue will be addressed in
future work.
In Eqs. (4.40) and (4.41) interpolated values of density and uid mass ux at the
interface are used. Even though these are discontinuous at the interface, interpolation
nds the average density, f = (v + l)=2, and mass ux, wf n = (wv n + wl n)=2.
Using Eq. (4.22) it is straightforward to show that
(f V , wf ) n = (v V , wv ) n = (lV , wl ) n :
(4.52)
Thus in this case the interpolated discontinuous quantities are identical to those on
either side of the interface and either can be used in Eqs.(4.40) and (4.41).
4.4 Results and Discussion
4.4.1 Comparison with an Exact Solution
The accuracy of the numerical method is tested by comparing numerical results
with the exact solution of a simple one-dimensional problem. The one-dimensional
problem consists of a heat ux, qw , applied to the bottom of a rigid wall at y = 0.
(This heat ux is made dimensionless by scaling it by Klv Lo=l cll.) The domain
contains a liquid 0 y 0:5 below its vapor 0:5 y 1. The top of the domain
at y = 1 remains open to allow for the vapor to exit due to uid expansion at the
111
interface. To illustrate relatively rapid interface motion the density ratio is set to
l=v = 2. (Good results for higher density ratios, up to l=v = 1000, are achieved,
however the interface velocity then becomes very small and the solution is controlled
by the large value of the vapor velocity.) All other material properties are equal. For
this calculation qw = 0:1; Re = 1; Pe = 1 and We = 1: There is no gravity, g = 0,
and the interface, initially at y = 0:5, remains planar.
The exact steady-state solutions for the interface velocity, uid velocities and
pressure jump across the interface are
V = ,v qw =l Pe ; ul = 0 ; uv = 1 , l V ; Pl , Pv = l , 1 V 2 : (4.53)
v
v
!
!
Figs. (4.1)-(4.4) compare results from the numerical solution for dierent grid resolutions to the exact solution. Fig. (4.1) compares the transient numerical calculation
for grid resolutions of 1010, 2020 and 4040 to the exact steady state value of the
interface velocity. (Note that the computations are performed in a two-dimensional
domain but the comparison with 1-D is valid since there are no variations exhibited
in the x,direction.) The results for all three resolutions are close and show the same
general behavior in reaching the exact steady-state value of -0.05. As expected the
results for the nest resolution are slightly more accurate. Fig. (4.2) makes the
same comparison for the uid velocity at time, t = 1:4. The results in the bulk
liquid and vapor are close for all the resolutions with the nest being slightly better.
However the results at the interface are of greater interest. The exact solution is
perfectly discontinuous while the numerical solutions approach this discontinuity as
the resolution increases. This behavior nicely demonstrates the convergence with
increasing grid resolution of the front tracking approach to modeling discontinuities
across an interface. The front tracking method inherently distributes the eects of
112
the interface smoothly to mesh points in a localized region near the interface. Thus
as the resolution increases these eects become sharper and more localized near the
interface. The nature of this nite thickness interface is clearly illustrated in a plot
of the density prole in Fig. (4.3). The numerical values of the density are calculated by Eq. (4.28). The interface, as depicted by the jump in density across it,
has a nite numerical thickness which decreases with increasing grid resolution. Fig.
(4.4) compares the results for the pressure jump at t = 1:4. The exact values of
the pressure in each phase are almost exactly matched by the numerical solution at
the highest resolution while the results for the lowest resolution exhibit more of a
discrepancy. The discontinuous jump in the pressure at the interface is accurately
simulated for the nest grid resolution. The spreading of the jump due to the nite
thickness of the interface is evident here as well.
4.4.2 Film Boiling
In these simulations the evolution of an unstable vapor layer below a liquid layer
which is below another vapor layer is followed. The computations are performed
in a rectangular 10 30 domain which is periodic in the x-direction. To allow
for vaporization uid is allowed to exit at the top boundary where the pressure is
specied to be zero. The temperature eld is initially zero everywhere with a heat
ux, qw , applied to the rigid bottom wall.
For the lm boiling problem, the length and velocity scales used to form the
dimensionless variables are, l = (2l =G2l )1=3 and Uo = (lG=l )1=3 respectively. With
this choice of length and velocity scales, Re = 1 and Fr = 1. The Weber number
becomes We = Mo1=3 where the Morton number is Mo = 4l G= 3 l and the Peclet
number becomes Pe = Pr where the Prandtl number is Pr = lcl=kl . In addition
113
the Nusselt number, Nu = Klqw =Kv Tw , is formed using the nondimensional wall
heat ux, qw , and wall temperature, Tw . Note that the Nusselt number is not set
beforehand but can change during the calculation depending on the value of the wall
temperature, Tw .
The lower of the two interfaces is given an initial shape described by
yf = yc + f cos (2nf xf =Hx )
(4.54)
where yc, f , nf and Hx are the average initial interface height, perturbation amplitude, perturbation mode and domain width respectively. The upper interface is
initially at.
The lm boiling simulations in Figs. (4.5) and (4.7) were run with the following
parameters
l = 10; l = 10; Kl = 10; cl = 1;
v
v
Kv
cv
qw = 0:5; Pr = 1; Mo = 1; = 0:002; # = 0:002;
yc = 5; f = ,1; nf = 1 :
The situation is now more complex than in the one-dimensional problem due to the
unstable dynamics of the interface in lm boiling.
In Fig. (4.5) the simulation is performed with a 50 100 grid resolution and is
shown for three dierent times. The interfaces are plotted as the solid white lines
while the arrows represent velocity vectors and are plotted only at every other grid
point. The temperatures are shown as shades of gray where the hottest regions
are near the bottom wall and the coolest regions are in the liquid which remains
nearly isothermal. In the rst frame of the gure the liquid vapor interface begins to
exhibit a Rayleigh-Taylor instability with the formation of counterrotating vortices.
Cold liquid is forced down toward the bottom wall and hot vapor is pushed up into
114
the forming bubble. The interface then pinches together and in the last frame the
separated bubble rises toward the upper interface carrying with it some of the hot
vapor from the heated wall.
To illustrate the variation of the pressure in the domain, in Fig. (4.6) a threedimensional view of the pressure eld at t = 10 is plotted. This calculation is the
same as in Fig. (4.5) except with twice the resolution. This gure only shows the
grid lines for every other grid point. The pressure rises toward the bottom of the box
(away from the viewer) due to hydrostatic forces. The discontinuity in the pressure
gradient at the upper liquid vapor interface is due to the increase in density from
vapor to liquid. The surface tension on the curved lower interface causes the pressure
to rise across regions of positive interface curvature and to drop across regions of
negative interface curvature.
The validity of these results is tested by comparing four dierent grid resolutions
in Fig. (4.7) at t = 10. Note that these results show the eect of time as well as grid
resolution since the maximum time step decreases with grid resolution. The 12 24
grid is clearly underresolved. The interface shapes for the two nest resolutions are
very close to one another and deviate only slightly at the crest and trough of the lower
interface. For these values of the physical parameters the 50 100 and 100 300
grids provide a nearly converged solution. Consistent with the study by LeVeque
and Li [136] linear convergence is observed in the dendritic and alloy solidication
results in Chapters II and III and the results in Fig. (4.7) also appear to exhibit
linear convergence.
Vapor generation due to vaporization at the liquid-vapor interface is clearly
demonstrated in the simulation shown in Fig. (4.8) where the density ratio and
the wall heat ux are now higher. The parameters are the same as in the previous
115
gure except
l = 100; l = 40; Kl = 20;
v
v
Kv
qw = 10; Mo = 8000 :
The amount of vapor in the growing bubble near the bottom wall is continually
increasing due to vaporization of liquid at the lower interface. Since the low density
vapor takes up more volume, the uid above it is pushed upwards. Note that there is
no vaporization at the upper interface. This interface simply moves passively upward
with the uid on either side of it. The fact that the upper interface does move upward
indicates that mass transfer is taking place across the lower interface.
In Fig. (4.9) heat transfer results from the two-dimensional simulations are compared against a correlation by Klimenko [163] on a plot of the Nusselt vs. Rayleigh
numbers. The Rayleigh number is Ra = 83 [Mov (l=v , 1)],1=2 Prv where Mov
and Prv are based on the vapor phase properties. Klimenko found that his correlation holds within 25% for many dierent uids. The open circles represent the
numerical results for three dierent runs at three dierent Rayleigh numbers. For
each of the three runs the average Nusselt number along the heated bottom wall
is plotted at dierent times. Thus the range of values for one run represents the
increasing and decreasing heat transfer from the wall as the height of the vapor
layer adjacent to the wall decreases or increases with time. The experimental results
would naturally give values averaged over the heated area as well as averaged in
time. The two-dimensional numerical results are consistently lower than the values
from three-dimensional experiments. However lower heat transfer in two dimensions
would be expected for several reasons. In three-dimensions the heat ow is not conned to a plane. There are also more bubbles rising from many various points on
a heated surface. On average the height of the vapor layer above the heated wall
116
would be lower and thus the heat transfer would be higher in three-dimensions than
in two-dimensions.
4.4.3 Rapid Evaporation
Next, results from two-dimensional simulations of evaporation from a superheated liquid under microgravity conditions are presented. For the rapid evaporation problem, the velocity scale used to form the dimensionless variables is,
Uo = Kl =lcll. Note that with this choice of velocity scale the Peclet number is
Pe = 1 and the Reynolds number becomes, Re = 1=Pr. In addition the Jakob number, Ja = lcl (T1 , Tv ) =v Lo , is the ratio of sensible to latent heat and is used here
to set the initial liquid superheat. The computations are performed with a 100 100
grid in a 1 1 domain which is periodic in the x,direction. The bottom wall is rigid
and adiabatic. To allow for vaporization uid is allowed to exit at the top boundary
where the pressure is specied to be zero. The temperature eld is initially set to
Ja = 1 everywhere.
In the simulation shown in Fig. (4.10) a layer of liquid rests below a layer of
vapor and the liquid-vapor interface is given an initial shape described by Eq. (4.54)
with
yc = 0:9; f = ,0:01; nf = 1 :
The other parameters are
l = 2; l = 1; Kl = 10; cl = 1;
v
v
Kv
cv
Pr = 1; We = 20; g = 0 = 0:001; # = 0:002;
In Fig. (4.10) the nearly horizontal solid white line is the liquid vapor interface
at time t = 0 and the highly convoluted line is the interface at t = 0:028. The
117
temperatures are shown as shades of gray at t = 0:028. White indicates the hottest
temperature, T = 1, and black is the coolest, T = 0. As the interface evaporates
and moves downward, small scale instabilities form. The larger interface length that
these instabilities present to the superheated liquid leads to a greater evaporation rate
which in turn magnies the protuberances. Eventually, the highly unstable interface
protrudes deep into the superheated liquid. Note that the two most convoluted arms
at either side of the box descend further into the liquid than the relatively smooth
arms nearer the center.
A plot of interface length vs. time in Fig. (4.11) illustrates the growth of the
interface instability which leads to explosive boiling. The interface length remains
nearly constant up to about t = 0:01 when a small amplitude instability sets in. The
growth of this instability is fairly constant until about t = 0:022. After this time the
interface length rapidly increases leading to rapid evaporation and explosive growth.
In the simulation shown in Figs. (4.12)-(4.15) an initially nearly circular interface
grows during rapid evaporation from a superheated liquid in microgravity. The
computations are performed with a 100 100 grid in a 1 1 domain which is periodic
in the x,direction. The initial conditions and the nondimensional parameters in this
case are the same as in Fig. (4.10) except
l = 10; l = 40; Kl = 20; cl = 1 :
v
v
Kv
cv
Depending on the particular uid and conditions these values are realistic. For
comparison, the properties of saturated uids are given in [171]. For cryogenic uids
such as hydrogen and oxygen, the values of the liquid to vapor density ratios are
roughly 54 and 253 respectively at a pressure of 1 atm. While at about 9 atm these
ratios are approximately 4 and 28 for hydrogen and oxygen respectively. These values
118
are typical of cryogenic uids and refrigerants while for water the ratio is roughly an
order of magnitude higher, 1600 and 147 at 1 atm and 12 atm respectively.
At 1 atm the viscosity ratio of liquid to vapor for these substances is in the range
of 20 to 30, the thermal conductivity ratio in the range 7 to 30 and the specic heat
ratio in the range 0.8 to 2. The Prandtl number, Pr, ranges from about 1 to 2
and for a Weber number of We = 20 the length scale of the instabilities is roughly
l = 10,4 m.
The initial interface is shown in the center of Fig. (4.12). In order to trigger
unstable growth, this initial interface is slightly perturbed from a circular shape and
is given by
xf = 0:5 + R cos ; yf = 0:5 + R sin (4.55)
where is measured counterclockwise from the x,axis and
R = 0:05 , 0:001 cos (8) :
(4.56)
This produces a symmetrically perturbed circle with eight lobes. The same gure also
shows the interface at times t = 0.03, 0.06, 0.09 and 0.12 as these eight lobes grow and
the interface develops into a highly convoluted shape. Fig. (4.13) shows the interface
at t = 0:03 along with contours of temperature. The thick line is the interface, the
outermost temperature contour is T = 0:877 and the innermost temperature contour
is T = 0:0148. Initially both the liquid and vapor were at a temperature of T = 1.
Now at t = 0:03 the vapor is at nearly zero temperature. As the bubble grows, the
energy in the superheated liquid is converted to latent heat. Thus the superheat
in the entire domain eventually becomes depleted, the temperature gradient in the
liquid at the interface drops and the bubble growth slows. This can be seen in Fig.
(4.12) as the expansion of the interface slows at later times. Fig. (4.14) shows the
119
interface at t = 0:03 along with the velocity eld. The velocity vectors indicate that
there is a general ow of liquid outward, away from the expanding bubble. The ow
in the vapor bubble is more complex due to the inow of vapor from the interface.
The velocity vectors in the core of the bubble show a general upward motion of the
entire bubble. This upward motion is consistent with the constraints of the rigid
bottom wall and the outward ow of ambient liquid at the top boundary. Note the
discontinuity of the uid velocity at the liquid-vapor interface due to the transfer of
mass across the interface. As the interface grows it remains nearly symmetric. Since
no symmetry requirements are imposed on the interface during the calculation this
result indicates that the interfacial features are suciently resolved.
The interface behavior and instability mechanism is similar to the previous simulation in Fig. (4.10). As the bubble grows outwards, increased evaporation in the
regions of the small initial protuberances cause unstable growth. This in turn results in a further increase in surface length and rapid evaporation. Eventually the
superheat in the liquid is depleted and the evaporation slows. However, even after
the period of energetic growth is over, the bubble continues to deform and eventually
begins to break apart. Fig (4.15) shows the interface at a later time, t = 0:15, after
the top two lobes of the interface have broken o from the main body of the bubble.
4.5 Conclusions
The single-eld, mathematical formulation of the Navier-Stokes and energy equations is described for the liquid-vapor phase change problem. A front tracking/nite
dierence solution technique is developed that includes the eects of interphase mass
transfer, latent heat, surface tension and unequal material properties between liquid
and vapor phases. The method is validated through comparison with an exact one-
120
dimensional solution and by grid resolution studies. Results from two-dimensional
simulations of lm boiling demonstrate the ability of the method to easily handle
large interface deformations and topology change. Heat transfer results are in good
agreement with a correlation of experimental results within the limitations of the
two-dimensional calculations. The simulations of rapid evaporation in superheated
liquids show the formation of highly convoluted unstable interfaces and enhancement
of evaporation leading to explosive growth. The simulations in Figs. (4.10)-(4.15)
were motivated by the observations of interface instability in microgravity boiling experiments [2, 3]. Although the conditions are not exactly the same, the simulations
show the qualitatively correct features and mechanisms of interface instability in microgravity boiling. Further work is underway to develop a quantitative comparison
with the microgravity boiling experiments.
121
0
Exact Steady State Solution
−0.01
n=10
20
40
Interface Velocity
−0.02
−0.03
−0.04
−0.05
−0.06
0
0.5
1
1.5
Time
Figure 4.1: Comparison of exact and numerical interface velocity for one-dimensional
boiling. The transient numerical calculation for grid resolutions of 10 10, 20 20 and 40 40 is compared to the exact steady state value of
the interface velocity. The results for all three resolutions are close and
show the same general behavior in reaching the exact steady-state value
of -0.05. As expected the results for the nest resolution are slightly
more accurate. l=v = 2; l=v = 1; Kl=Kv = 1; cl=cv = 1; qw =
0:1; Re = 1; g = 0; Pe = 1; We = 1; = 0:002; # = 0:002
122
0.06
Velocity
0.04
0.02
Exact Solution
n=10
20
0
0
40
0.1
0.2
0.3
0.4
0.5
Y
0.6
0.7
0.8
0.9
1
Figure 4.2: Comparison of exact and numerical uid velocities for one-dimensional
boiling at time, t = 1:4. The results in the bulk liquid and vapor are
close for all the resolutions with the nest being slightly better. At the
interface the exact solution is perfectly discontinuous while the numerical
solutions approach this discontinuity as the resolution increases. The
front tracking method inherently distributes the eects of the interface
smoothly to mesh points in a localized region near the interface. Thus as
the resolution increases these eects become sharper and more localized
near the interface.
123
1.1
Exact Solution
1
n=10
20
Density
0.9
40
0.8
0.7
0.6
0.5
0.4
0
0.1
0.2
0.3
0.4
0.5
Y
0.6
0.7
0.8
0.9
1
Figure 4.3: The nite numerical thickness of the interface is clearly illustrated in the
density prole. The numerical values of the density are calculated by
Eq. (4.28). The interface, as depicted by the jump in density across
it, has a nite numerical thickness which decreases with increasing grid
resolution.
124
3
x 10
−3
Exact Solution
2.5
n=10
20
Pressure
2
40
1.5
1
0.5
0
−0.5
0
0.1
0.2
0.3
0.4
0.5
Y
0.6
0.7
0.8
0.9
1
Figure 4.4: Comparison of exact and numerical pressure for one-dimensional boiling
at time, t = 1:4. The exact values of the pressure in each phase are almost exactly matched by the numerical solution at the highest resolution
while the results for the lowest resolution exhibit more of a discrepancy.
The discontinuous jump in the pressure at the interface is accurately simulated for the nest grid resolution. The spreading of the jump due to the
nite thickness of the interface is evident particularly at low resolution.
125
Figure 4.5: A lm boiling simulation at three dierent times. The solid white lines
are the liquid-vapor interfaces, the arrows represent velocity vectors and
are plotted only at every other grid point. The temperatures are shown
as shades of gray where the hottest regions are near the bottom wall and
the coolest regions are in the liquid which remains nearly isothermal. A
Rayleigh-Taylor instability forms with subsequent pinch-o and rise of
a vapor bubble. The bubble carries heated vapor up into the ambient
liquid. The simulation is performed on a 50 150 grid with l=v =
10; l =v = 10; Kl =Kv = 10; cl=cv = 1; qw = 0:5; Pr = 1; Mo =
1; = 0:002; # = 0:002
126
15
10
5
0
0
5
10
10
15
8
20
6
4
25
2
30
0
Figure 4.6: The variation of the pressure in the domain at t = 10. This calculation
is the same as in Fig (4.5) except with twice the resolution. This gure
only shows the grid lines for every other grid point. The pressure rises
due to hydrostatic forces toward the bottom of the box (away from the
viewer). The discontinuity in the pressure gradient at the upper liquid
vapor interface is due to the increase in density from vapor to liquid. The
surface tension on the curved lower interface causes the pressure to rise
across regions of positive interface curvature and to drop across regions
of negative interface curvature.
127
30
12 x 24
25
25 x 75
50 x 150
100 x 300
Y
20
15
10
5
0
0
5
X
10
Figure 4.7: A grid resolution study for lm boiling. Interface shapes for four dierent
grid resolutions are compared at time, t = 10. The 12 24 grid is clearly
underresolved. The interface shapes for the two nest resolutions are
very close to one another and deviate only slightly at the crest and trough
of the lower interface. For these values of the physical parameters the
50 100 and 100 300 grids provide a nearly converged solution. The
calculation is in a 10 30 rectangular domain and the other parameters
are the same as in Fig. (4.5)
128
Figure 4.8: A lm boiling simulation for three dierent times with the density of the
liquid 100 times that of the vapor. The amount of vapor in the growing
bubble near the bottom wall increases due to vaporization of liquid at the
lower interface. Since the low density vapor takes up more volume, the
uid above it is pushed upwards and the upper interface moves passively
upward with the uid. The parameters are the same as in Fig. (4.5)
except l=v = 100; l=v = 40; Kl =Kv = 20; qw = 10; Mo = 8000 .
129
2
10
Klimenko Correlation
25% Experimental Data Scatter
2D Numerical Results
1
Nu
10
0
10
−1
10
10
1
2
10
3
10
Ra
4
10
5
10
Figure 4.9: Comparison of heat transfer results from two-dimensional numerical simulations against a correlation by Klimenko [163]. For each of the three
numerical runs at three dierent Rayleigh numbers the average Nusselt
number along the heated bottom wall is plotted at dierent times. Twodimensional simulations are in good agreement but consistently lower
than results from three-dimensional experiments. Higher values in threedimensions are expected since heat ow is not conned to a plane and
on average the height of the vapor layer above the heated wall would
be lower and thus the heat transfer would be higher in three-dimensions
than in two-dimensions.
130
Figure 4.10: Rapid evaporation from an initially nearly planar interface in microgravity. The solid white lines are the liquid vapor interfaces at times
t = 0 and t = 0:028. The temperatures are shown as shades of gray
at t = 0:028. White indicates the hottest temperature, T = 1, and
black is the coolest, T = 0. The highly unstable interface protrudes
downward into the superheated liquid. Evaporation is enhanced in areas where the interface convolution and thus interface length is greatest.
The computation is performed on a 100 100 grid in a 1 1 domain
with l=v = 2; l=v = 1; Kl =Kv = 10; cl=cv = 1; Pr = 1; We =
20; g = 0 = 0:001; # = 0:002
131
10
9
8
Interface Length
7
6
5
4
3
2
1
0
0
0.005
0.01
0.015
Time
0.02
0.025
0.03
Figure 4.11: A plot of interface length vs. time illustrates the growth of the interface
instability which leads to explosive boiling. The interface length remains
nearly constant up to about t = 0:01 when a small amplitude instability
sets in. The growth of this instability is fairly constant until about
t = 0:022. After this time the interface length rapidly increases leading
to rapid evaporation and explosive growth.
132
1
0.9
0.8
0.7
Y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
X
Figure 4.12: The growth of interface instabilities on a bubble growing in a superheated liquid in microgravity. The initial interface is shown as the
nearly circular shape in the center of the gure. Also shown the are the
interface at times t = 0.03, 0.06, 0.09 and 0.12 as the interface develops
into a highly convoluted shape. The computation is performed on a
100 100 grid in a 1 1 domain. The parameters here are the same as
in Fig. (4.10) except l =v = 10; l=v = 40; Kl=Kv = 20 :
133
1
0.9
0.8
0.7
Y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
X
Figure 4.13: The interface and temperature contours at t = 0:03. The thick line is
the interface, the outermost temperature contour is T = 0:877 and the
innermost temperature contour is T = 0:0148. As the bubble grows,
the energy in the superheated liquid is converted to latent heat. Thus
the superheat eventually becomes depleted, the temperature gradient
in the liquid at the interface drops and the bubble growth slows. This
can be seen in Fig. (4.12) as the expansion of the interface slows at
later times.
134
1
0.9
0.8
0.7
Y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
X
Figure 4.14: The interface and velocity eld at t = 0:03. The velocity vectors indicate
that there is a general ow of liquid outward, away from the expanding
bubble. The ow in the vapor bubble is more complex due to the
inow of vapor from the interface. The velocity vectors in the core of
the bubble show a general upward motion of the entire bubble. This
upward motion is consistent with the constraints of the rigid bottom
wall and the outward ow of ambient liquid at the top boundary. Note
the discontinuity of the uid velocity at the liquid-vapor interface due
to the transfer of mass across the interface.
135
1
0.9
0.8
0.7
Y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
X
Figure 4.15: Bubble breakup. The interface is shown at time, t = 0:15, after the top
two lobes of the interface have broken o from the main body.
CHAPTER V
SUMMARY
5.1 Present Work
The objective of the present work has been to develop a predictive capability
for phase change in multiphase, multicomponent systems. To do this computational
methodologies have been developed for phase change in three cases: pure material
solidication (Chapter II), alloy solidication (Chapter III) and liquid-vapor phase
change with uid ow (Chapter IV).
For each case:
A comprehensive single-eld mathematical model of the physical system is
presented which includes large interface deformations, topology change, latent
heat, interfacial anisotropy and discontinuities in material properties between
the phases.
The implementation of a computational method for its solution based on a
front tracking/nite dierence technique is developed and described.
This method is validated through comparisons with theories or exact solutions
and also through grid resolution studies.
136
137
The full capability of the method is demonstrated in simulations of complex
unstable interfaces where multiple interacting physical eects come into play.
The specic contributions to the state of the art made in this work are:
Chapter II: Pure Material Solidication.
The development of an alternative method for time-dependent, two-dimensional
dendritic solidication of pure materials. The ability of the method to handle
the general case of unequal material properties allowed the exploration and
identication of a previously unreported increase in instability associated with
unequal specic heats between the liquid and solid phases.
Chapter III: Alloy Solidication.
The development of a numerical method for time-dependent, two-dimensional
solidication of dilute binary alloys. This appears to be the rst method which
solves the coupled, solute and energy equations including the eects of large
interface deformations, topology change, latent heat, anisotropic capillarity and
interface kinetics and discontinuities in material properties between the liquid
and solid phases. The major contribution is that latent heat is not neglected
and a known temperature or an isothermal system is not assumed.
Chapter IV: Liquid-Vapor Phase Change with Fluid Flow.
Development of a method for time-dependent, two-dimensional liquid-vapor
phase change with uid ow. This appears to be the only method available
at present which takes into account mass, momentum and energy transport
with interphase mass transfer while allowing large interface deformations and
topology change. It also accounts for latent heat, surface tension and unequal
material properties between the liquid and vapor phase.
138
5.2 Recommendations for Future Work
5.2.1 Numerical Enhancements
Here some unresolved issues and several ideas for enhancements to the present
capability of the numerical methodology developed in this work are identied.
In all of the simulations an explicit time integration scheme is used. While this
is initially the easiest method to use, in some cases the method is restricted by
numerical stability considerations to small time steps and thus long computation
times. For future computational studies, it would be practical to incorporate implicit
time integration into the numerical methodology. This would allow greatly reduced
computation times, particularly for the alloy solidication problem where there is
usually a great discrepancy in the time scales of thermal and solutal diusion.
Parallel computers are becoming increasingly more powerful and more user friendly.
The recent standardization of communication subroutine libraries used for programming on parallel computers will allow portability of programs across platforms. The
addition of usable, massively parallel computers to the computational scientist's toolbox promises to provide a quantum leap in modeling capability and quality. In the
near future high resolution calculations with hundreds of millions of grid points and
higher will be routine. Future implementations of the numerical methodology should
certainly include parallel machines.
In the liquid-vapor phase change problem, the correct pressure dependence of the
vaporization temperature at the phase change interface is still an unresolved physical
issue. In this work it has been assumed that the interface is in thermal equilibrium
but not necessarily thermodynamic equilibrium (see section 4.2). There appear to
be a number of other choices for the interface temperature condition. Using the
method developed here, these choices could now be explored numerically in an eort
139
to help resolve this uncertainty. In addition, since the pressure is discontinuous across
the interface, diculty is encountered in numerically determining its interfacial value
during the computation. In the alloy problem, a similar discontinuity in the interface
concentration is avoided by using a simple transformation. A similar approach for
the discontinuous interface pressure in the liquid-vapor phase change problem should
be investigated.
The incorporation of a temperature dependent surface tension is relatively straightforward [135] and could easily be done to study thermocapillary ows with phase
change.
The extension of the computational methodology for phase change described
here, to three-dimensional simulations promises to yield dramatic new insight and
understanding to phase change processes. The main barriers are the computational
requirements and the necessity to track an evolving two-dimensional surface through
three-dimensional space. Surface-tracking has already successfully been implemented
[64{66] and with the advancement of parallel computers and algorithms, large scale
three-dimensional simulations are becoming more common. However, in order to
resolve detailed, three-dimensional solidication microstructures with a reasonable
computational expense, it will likely be necessary to incorporate adaptive meshing
to allow ner resolution near the interface.
In engineering applications, phase change takes places in devices or environments
with a much more complex geometry than the simple rectangular domains that have
been used here. Heat exchangers consist of bundles of nned tubes and castings for
complex components usually involve complicated internal and external geometrical
features. In order to model these geometries it would be advantageous to compute on unstructured grids using nite element or nite volume methods. Since
140
the front tracking routines are modular and essentially independent of the particular method used to solve the governing equations on the underlying stationary grid,
these routines could easily be implemented alongside nite elements or nite volumes
on unstructured stationary grids.
5.2.2 Applications
Compared to pure, single phase materials and uids, the numerical investigation
of multiphase, multicomponent systems with phase change is a vast and relatively
unexplored frontier [75{77]. In this thesis a numerical capability to begin exploring
this frontier has been developed.
In order to be able to predict the desired outcome of a process or properties of
a nal product it is necessary to accurately model complex interacting phenomena.
This is especially true in ows with phase change where none of the interactions
among uid ow, surface tension, heat transport and phase change can be ignored.
Engineering predictions of large scale multiphase ows with phase change typically
use some form of averaged, multiuid model of the governing continuum equations in
each phase along with empirical ow regime maps. The averaging procedure results
in unknown interfacial closure terms which are usually empirically determined and
often poorly measurable. In describing recent advancements in the development of
two-uid models for phase change, Jones and Lahey [76] write,
\What is needed in order to realize the full potential of these multidimensional
models is the development of physically-based interfacial closure laws and jump conditions for each ow regime of interest so that we do not repeat the mistakes of the
past which resulted in highly empirical closure laws, with no real improvement in
prediction capability. The new constitutive relations must be obtained from, and
141
assessed against, suitable separate-eects data and/or exact analytical results, and
should have few, if any, tunable parameters."
The numerical method developed here can directly help to determine these closure
laws by providing the statistical information necessary from suitably large scale simulations.
On the microscale, research is underway to determine how vaporization and condensation of tiny bubbles can be used to drive miniature heat engines in microdevices
or to act as switches to control micromachine elements. Numerical computations can
be directly utilized in the design of these systems and the prediction of heat and mass
transfer rates, bubble dynamics, uid forces and operating eciency. Along similar
lines, investigations by Asai [172] were motivated by the use of vapor bubbles to
generate pressure pulses in thermal printers.
Solidication processes often involve some degree of uid ow in the melt prior
to solidication. In order to control residual stresses and ensure consistent and desirable physical properties of manufactured components, the detailed interaction of
solidication with uid ow must be understood. Thermal convection in Bridgman
directional solidication techniques [173] inuences the uniformity of the solid microstructure. In Czochralski techniques [174] a rotating rod is carefully pulled from a
crucible containing melt. Solidication continuously occurs at the end of the rod as
it is lifted out of the crucible. Lifting and rotation rates must be carefully balanced
with capillary, gravitational and viscous uid forces as phase change takes place.
Rapid solidication processes such as spray casting [142] involve atomizing metal
into a spray which partially solidies in ight prior to deposition on the desired
substrate. By suitably manipulating the motion of this substrate, a near-net shape
product with a uniform, ne grained microstructure can be fabricated directly from
142
the melt. One of the challenges to producing consistent part shapes and properties includes the detailed understanding of the inuence of the unsolidied liquid in
the interstices of the previously solidied drops on the time required for complete
solidication.
In microfabrication [175], careful low-speed deposition of drops is used to build
up three-dimensional structures. Using a simple model of solidication, the front
tracking method has been applied in preliminary simulations of this process [176].
In rheocasting [177], a semi-solid charge of slurry is injected into a die where it
deforms to the internal die shape and subsequently solidies. Since the initial charge
has begun to solidify prior to injection the eects of uid ow during injection on
the simultaneously evolving microstructure is poorly understood. Direct numerical
simulations could be used to investigate the inuence of uid ow and capillary
forces during injection or shrinkage upon solidication on the desired cast shape and
microstructure.
In combustion science the propagation of premixed ames can, in some instances,
be treated very similarly to the liquid-vapor phase change process where now the
gas expands upon combustion and heat is released at the ame interface due to
chemical reaction. Qian et al. [178] have adopted the method described in section
IV to perform isothermal calculations of combustion in premixed ames. In their
simulations only the hydrodynamics needs to be considered since it is assumed that
the ame speed relative to the unburnt gas is known. It appears that with added
capabilities to the numerical method developed in this thesis for liquid-vapor phase
change, the basic method could also be used in the calculation of more complex
combustion problems where the ame speed is unknown and the transport of energy
and chemical species must be accounted for.
143
The petroleum and chemical industries often encounter multicomponent, multiphase processes in many varieties of industrial reactors. In a multiphase reactor
such as a slurry reactor [179], reactant gas is bubbled through a liquid solution containing solid catalyst particles. The reactants in the gas diuse from the gas to
the liquid and in turn to the catalyst where they react to synthesize new chemicals.
This type of reactor is used in the synthesis of methane and distillation of oil, for
example. Reaction rates which govern the overall eciency of the reactor, are almost
always determined by highly empirical correlations. Numerical simulations oer the
potential to contribute signicantly to reactor design and eciency.
The biotechnology industry uses similar reactor techniques in the biosynthesis of
a growing list of commercially important products such as insulin, antibiotics and
polymers. From 1990 to 2000 it is projected that the market for products produced
by biosynthesis will increase 60 times to $17 billion [180]. In bioreactors the solid
catalyst is replaced by living cells to which nutrients must be transported and from
which excreted waste must be removed. The scale-up of bioreactors to industrial
proportions poses signicant problems since uid shear stresses during mixing of the
cells with nutrients can easily destroy the fragile cells. Important contributions stand
to be made in the numerical investigation of the eects of phase change and uid
and surface forces, on the transport of energy and chemical species.
The uncontrolled release of dissolved gases from supersaturated liquids can lead to
destructive accidents in reneries or chemical plants and is also thought to contribute
to violent volcanic eruptions. Studying the combined mechanisms of boiling and
degassing could lead to considerable new insight.
The capability to study multiphase, multicomponent processes could be added
to the numerical methodology described in this work in a straightforward manner
144
with the addition of equations governing species transport. It appears that simulations of multicomponent, multiphase ows are within reach and are among the next
challenges for computational researchers.
Only some of the many applications for computations of phase change have been
discussed here. The list is nearly inexhaustible and encompasses many diverse elds
of engineering and science. Future advancements will depend heavily on maintaining
a broad, interdisciplinary vision of opportunities and challenges for computational
physics.
APPENDIX
145
146
APPENDIX A
THE FRONT-TRACKING METHOD
The front tracking concepts described in this chapter are common to all three
phase change problems considered in this work. These concepts consist of:
1. An interface representation to calculate local interface quantities such as curvature, normal and tangent.
2. Interface reconstruction to allow for deformation and topology change.
3. Distribution and interpolation to transfer information between the moving front
and the stationary grid.
4. The numerical construction of the indicator function, I (x; t), to advect the
material property elds.
A.1 Interface Representation
To explicitly mark the interface separate, non-stationary computational points
are connected to form a one-dimensional front which lies within the two-dimensional
stationary mesh. The interface is represented by the vector parametric equation
R(u) = g(u)i + h(u)j :
(A.1)
147
One can nd the normal, tangent and curvature at any point on the interface using
the formulae
n = ,h i + g j ;
0
0
0
0
(A.2)
g2 +h2
t = g i2+ h j2 ;
(A.3)
g +h
= g h2 , g2 h3=2 ;
(A.4)
g +h
where the prime denotes dierentiation with respect to the parameter u and i and j
q
0
0
q
0
0
0
00
0
00
0
0
are unit vectors in the x and y directions respectively. In practice since the positions
of the interface points are known, formulae for the component functions g and h
are developed by tting an nth order Lagrange polynomial through a set of n + 1
adjacent points
gn (u) =
n
X
i=0
Li(u)xf (ui) ;
where
Li(u) =
hn (u) =
n
X
i=0
Li (u)yf (ui)
u , uj
j =0 ui , uj
n
Y
(A.5)
(A.6)
j 6=i
and (xf (ui); yf (ui)) are the coordinates of the discrete interface points. (Note that in
order to maintain the sign convention for normal and curvature, the sequence of adjacent points (xf (ui); yf (ui)) should be chosen so that as the parameter u increases,
the normal vector points into the solid/vapor region and the curvature is positive
when the center of curvature lies in the solid/vapor region.) A fourth order polynomial is constructed through ve successive interface points (xf (ui); yf (ui)); i = 0; 4.
Choosing the parameterization, ui = i; i = 0; 4, the normal, tangent and curvature
at the point (xf (u2); yf (u2)) are found using Eqs. (A.2)-(A.4) with
g4(u2) = (xf (u0) , 8xf (u1) + 8xf (u3) , xf (u4)) =12 ;
0
(A.7)
148
h4(u2) = (yf (u0) , 8yf (u1) + 8yf (u3) , yf (u4)) =12 ;
0
(A.8)
g4 (u2) = (,xf (u0) + 16xf (u1) , 30xf (u2) + 16xf (u3) , xf (u4)) =12 ;
(A.9)
h4 (u2) = (,yf (u0) + 16yf (u1) , 30yf (u2) + 16yf (u3) , yf (u4)) =12 :
(A.10)
00
00
A.2 Interface Reconstruction
The interface deforms greatly in the simulations and it is necessary to add and
delete interface points during the course of the calculation such that the distance between adjacent points, d, is maintained on the order of the stationary grid spacing, h.
For the simulations in this work 0:4 < d=h < 1:6 is used. To accommodate topology
changes, interfaces are allowed to reconnect when either parts of the same interface
or parts of two separate interfaces come close together. The instantaneous change in
topology is, of course, only an approximation of what happens in reality. Since it is
not well known at what distance the interfaces will coalesce when brought together
and distances at such a small scale are not resolved, the interface is articially reconnected when two points come closer than a small distance, p. This distance is chosen
rather arbitrarily for lack of a better physical model. But here the advantage of front
tracking is evident since the distance at which interfaces merge can be controlled and
the eect of varying p can be studied, unlike in interface capturing methods such as
the phase eld (see [19] for example) or level set [50, 51] methods where there is no
active control over topology changes. While the above modications to the interface
are a major task for three-dimensional simulations, here the interface is simply a
line and they are relatively straightforward. The interface points are connected by
forward and backward linked lists and interface restructuring is simply a matter of
resetting pointers.
149
A.3 Distribution and Interpolation
At each time step information must be passed between the moving Lagrangian
interface and the stationary Eulerian grid since the Lagrangian interface points,
xk , do not necessarily coincide with the Eulerian grid points, xij . The innitely
thin interface is approximated by a smooth distribution function that is used to
distribute sources at the interface (due to liberation/absorption of latent heat, q,
rejection/absorption of solute, s, mass transfer across the interface, m or surface
tension, f ) over grid points nearest the interface. In a similar manner, this function
is used to interpolate the eld variables from the stationary grid to the interface. In
this way, the front is given a nite thickness on the order of the mesh size to provide
stability and smoothness. There is also no numerical diusion since this thickness
remains constant for all time. Using latent heat, q, and temperature, T , as examples,
the interfacial sources, qk , can be distributed to the grid and the grid eld variables,
Tij , can be interpolated to the interface by the discretized summations
Qij =
Tk =
X
k
qk Fij (xk ) sk ;
(A.11)
h2Tij Fij (xk )
(A.12)
X
ij
where sk is the average of the straight line distances from the point k to the two
points on either side of k. Eq. (A.11) is the discretized form of Eq. (2.13) where
the Dirac function has been approximated by the distribution function, Fij . For
xk = (xk ; yk ) the distribution function suggested by Peskin [181] is used
Fij (xk ) = f (xk =h , i)h2f (yk =h , j )
(A.13)
150
where
8
>
>
>
>
>
>
>
<
jrj 1;
f (r) = 1=2 , f1 (2 , jrj) ; 1 < jrj < 2;
0;
jr j 2
f1(r);
>
>
>
>
>
>
>
:
and
(A.14)
q
3 , 2 jrj + 1 + 4 jrj , 4r2
f1(r) =
:
(A.15)
8
Peskin's [141] cosine distribution function was also tried and no discernible dierence
in the results was found.
A.4 Calculation of the Indicator Function
Discontinuous material properties can easily be accommodated through the numerical construction of an indicator function, I (x; t). The jump in the indicator
function across the interface is distributed to the grid points nearest to the interface
using Eq. (A.11). This generates a grid-gradient eld,
Z
G(x) = rI = A n (x , xf ) dA ;
(A.16)
which is zero except near the interface, and has a nite thickness. The divergence of
the gradient eld is found by numerical dierentiation, using second-order centered
dierences (r G). Thus the Laplacian of the indicator function is calculated and
this is again zero, except near the interface. To nd the indicator function everywhere
the Poisson equation
r2I = r G
(A.17)
is solved. The indicator function is constant within each material region, but has
a nite thickness transition zone around the interface and therefore approximates a
two-dimensional step-function. The primary advantage of this approach is that close
151
interfaces can interact in a natural way since the gradients simply add or cancel as
the grid distribution is constructed from the information carried by the tracked front.
Therefore, when two interfaces are close together the full inuence of the latent heat
from both interfaces is included in the heat equation. It should be noted that if the
material properties are equal in both the liquid and solid regions then there is no
need to construct the indicator function.
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