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UNIVERSITY OF NOVA GORICA
GRADUATE SCHOOL
CONTRIBUTION TO DEVELOPMENT OF MESHLESS
METHODS FOR FREE AND MOVING BOUNDARY
PROBLEMS
DISSERTATION
Nazia Talat
Mentor: Prof. Dr. Božidar Šarler
Nova Gorica, 2018
UNIVERZA V NOVI GORICI
FAKULTETA ZA PODIPLOMSKI ŠTUDIJ
PRISPEVEK K RAZVOJU BREZMREŽNIH METOD ZA
PROBLEME S PROSTIMI IN PREMIČNIMI MEJAMI
DISERTACIJA
Nazia Talat
Mentor: Prof. Dr. Božidar Šarler
Nova Gorica, 2018
UNIVERSITY OF NOVA GORICA
GRADUATE SCHOOL
Author Nazia Talat, Contribution to development of meshless methods
for free and moving boundary problems, Dissertation, (2018)
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Acknowledgements
First and foremost, I would like to express my profound gratitude to my mentor Prof.
Dr. Božidar Šarler for his support, patience, guidance and encouragement for
finishing the present dissertation. I would also like to thank the committee members,
Prof. Dr. Ching-Shyang Chen, Prof. Dr. Iztok Tiselj and Prof. Dr. Jurica Sorić for
their suggestions, remarks and comments.
I am particularly thankful to my colleagues Dr. Katarina Mramor, Dr. Qingguo Liu,
Dr. Umut Hanoglu, Grega Belšak, Tadej Dobravec, Dr. Rizwan Zahoor for unlimited
discussions, suggestions and ideas. I am deeply grateful to Dr. Boštjan Mavrič and
Vanja Hatić for their continuous help and guidance in the development of the pre-
existing numerical model. I am also thankful to Mrs. Tea Stibilj Nemec, Mrs. Saša
Badalič and Mrs. Vesna Mržek for their help in all academic administrative activities
and fruitful discussions during my leisure time.
I am specially thankful to Centre of Free Electron Laser (CFEL), Deutsches
Elektronen-Synchrotron (DESY), Hamburg for co-financing the present research in
the framework of the project “Innovative methods for imaging with the use of X-ray
Free Electron Laser (XFEL) and synchrotron sources” and Slovenian Research
Agency project J2-7384: Advanced modelling of liquid-solid systems with free and
moving boundaries and program group P2-0162 Transient two-phase flows. I am
grateful to Dr. Saša Bajt (Photon Science, DESY), Prof. Dr. Henry Chapman (CFEL-
DESY, University of Hamburg, Center for Ultrafast Imaging) for helpful technical
discussions.
I would like to acknowledge Institute of Metals and Technology, Ljubljana, for
providing me the computational facilities to perform numerical simulations.
Last, but not least, I would like to thank my family and friends for their support,
patience, understanding and most importantly their encouragements.
Contribution to Development of Meshless Methods for Free
and Moving Boundary Problems
Abstract
The purpose of this dissertation is the development of a novel, diffuse approximate
meshless method in connection with the phase field formulation for solution of free
and moving boundary problems. Free and moving boundary problems arise in a wide
variety of scientific and technological applications. The common feature of such
problems is the topological evolution of the interfaces between the phases, which
leads to the formation of various flow patterns and regimes that strongly depend on
the physical properties of the phases. Evolution of deformed interfaces remain
challenging from the physical and the computational points of view due to the strong
effects of surface tension and the discontinuities across the interfaces.
We deal with the numerical modelling and simulation of two-phase flow using
diffuse interface method, namely, phase field method in the present dissertation. Both
phases are considered to be Newtonian, incompressible and immiscible. The problem
is formulated with coupled Navier-Stokes and Cahn-Hilliard equations. Navier-
Stokes equations govern the flow of the two fluids and Cahn-Hilliard equation is
used for representation of the surface tension and to describe the evolution of the
interface. The diffuse approximate method is structured with second order
polynomial shape functions, Gaussian weights, local domain support and upwinding.
The pressure-velocity coupling is performed by an incremental pressure correction
scheme. The governing equations are solved in two-dimensions (2D) and in
axisymmetry by using explicit time discretization.
The performance of the method is tested on the well known 2D Rayleigh-Taylor
instability problem using three different physical models: (I) a model with large
density variation and surface tension, (II) a model with Boussinesq formulation for
small density variation, and (III) a model with phase field dependent density for
small density variation without the surface tension. The assessment of the method is
carried out based on the sensitivity analysis by using different values of the shape
parameter and the number of nodes in the local subdomain for the model (I).
Furthermore, the simulations are performed on three different node arrangements
64x256, 128x512 and 192x786 to demonstrate the node density convergence. The
effect of the Atwood number At = 0.01, 0.1, 0.3 and 0.5 on the height of the bubbles
and the spikes is carried out for the models without surface tension effect. The fine-
node arrangement results are compared with the previously published results
obtained by staggered Marker-And-Cell method and finite volume method results,
calculated by using open source finite volume method based computational fluid
dynamics code Gerris.
The meshless results demonstrate that the dynamics of Rayleigh-Taylor instability
can be efficiently evaluated by a combination of phase-field method and meshless
solution procedure. By comparing the results of all aforementioned models with the
reference results from literature, it is found that the surface tension significantly
changes the shape of the moving boundary between the fluids and that the largest
curvature appears on both left and right tail of the mushrooms. Furthermore, the
sensitivity study shows that when using nine nodes in local subdomain for shape
parameter c = 2.5 and 5, the results are almost the same, however for c = 10,
simulations show that the heavy front moves faster than the lighter one and a
significant bend of both left and right tails appears. When using different number of
nodes in local subdomain (e.g. eleven and thirteen) with c = 10, the results are similar
demonstrating that the simulations are not sensitive to the number of nodes in local
subdomain and are in close agreement with the finite volume results. The meshless
results are also verified by reproducing the moving boundary dynamics, consistent
with the previously published results: for an initially symmetric perturbation of the
interface, the symmetry of the heavy and light fronts of the Boussinesq model is
preserved for a long time. However, for the variable density model, the related
symmetry is lost despite the fact that the flow starts symmetrically.
The method is afterwards applied also in axisymmetry for a problem of two co-
flowing Newtonian, incompressible and immiscible fluids with different material
properties that yield dripping or jetting of the core fluid. An assessment of the novel
method is carried out based on the node density convergence in terms of calculated
dimensionless jet length. The meshless results are compared with the finite volume
results provided with an open source numerical toolbox OpenFOAM. A sensitivity
study of the various process parameters such as capillary number and viscosity ratio
is also studied and verified against the previously published results. It is found that
there is no significant difference in the calculated jet length for discretisations:
30x400, 45x600 and 60x800 and the results with node arrangement 30x400 are
reasonably accurate. The meshless results demonstrate excellent agreement with the
finite volume results in terms of drop size and temporal behavior of interphase
boundary. The meshless results are also in good agreement with the finite difference
results in terms of dimensionless limiting length and volume of the drop. The
combination of diffuse approximate method is also suitable for tackling the
axisymmetric forced-flow moving boundary two-phase flow problems.
This represented work deals with a pioneering attempt in solution of 2D Rayleigh-
Taylor instability and the axisymmetric forced-flow moving boundary two-phase
flow problems by a meshless solution of the phase field formulation. The results
show that the combination of diffuse approximate method and phase field method is
capable to handle free surface flows with large topological changes and provide a
valuable numerical tool for solving immiscible convective hydrodynamics.
Furthermore, extensive experiments are carried out using the combination of diffuse
approximate method and phase field method for the numerical simulation of gas
focused micro-jets formed by gas dynamic virtual micro-nozzles. After such
experiments, it is found that the current version of the method is not yet suitable for
gas focused micro jets due to very large density difference of liquid (water) and gas
(helium) and many very sharp edges of the nozzle. In the perspective, there is a need
to further explore a special treatment of sharp edges and pressure velocity coupling
scheme to handle very large density variations.
Keywords
Two-phase flow, free and moving boundaries, computational fluid dynamics, phase-
field formulation, 2D problems, axisymmetric problems, diffuse approximate
meshless method, Rayleigh-Taylor instability, Boussinesq approximation, variable
density and viscosity, flow focusing, dripping, jetting
Prispevek k razvoju brezmrežnih metod za probleme s
prostimi in premičnimi mejami
Povzetek
Namen te disertacije je razvoj nove, brezmrežne metode difuznih približkov v
povezavi s formulacijo faznega polja za probleme s prostimi in premikajočimi se
mejami. Takšni problemi se pojavijo pri široki množici znanstvenih in tehnoloških
problemov. Skupna lastnost teh problemov je topološki razvoj medfaznih mej, kar
vodi do oblikovanja raznovrstnih tokovnih vzorcev in režimov, ki so močno odvisni
od fizikalnih lastnosti prisotnih faz. Razvoj deformiranih mej predstavlja izziv iz
fizikalnih ter računskih vidikov zaradi močnega vpliva površinske napetosti ter
nezveznosti na obeh straneh mej.
V disertaciji obravnavamo numerično modeliranje in simulacijo dvofaznega toka z
uporabo metode difuzne meje, točneje, z metodo faznega polja. V delu privzamemo,
da sta obe fazi Newtonski, nestisljivi in, da se ne mešata. Problem je opisan s
povezanimi Navier-Stokesovimi in Cahn-Hilliardovo enačbo. Navier-Stokesove
enačbe opisujejo tok obeh tekočin, Cahn-Hilliardova enačba pa je potrebna za
izračun prispevka površinske napetosti ter časovnega razvoja medfazne meje.
Metoda difuzni približkov je strukturirana na podlagi polinomskih oblikovnih funkcij
drugega reda, Gaussovimi utežmi, lokane podporne domene ter privetrne sheme.
Hitrostno-tlačna sklopitev je narejena na podlagi inkrementalne sheme tlačnega
popravka. Sistem enačb je rešen v dveh dimenzijah (2D) ter v osni simetriji z
uporabo eksplicitne časovne diskretizacije.
Obnašanje metode je preverjeno na dobro poznanem dvodimenzionalnem problemu
Rayleigh-Taylorjeve nestabilnosti z uporabo treh različnih fizikalnih modelov: (I)
model z velikim variacijami gostote in površinsko napetostjo, (II) model z
Boussinesqovo formulacijo za majhne variacije gostote ter (III) model za majhne
variacije gostote z gostoto odvisno od faznega polja a brez vpliva površinske
napetosti. Metoda je ovrednotena na osnovi občutljivostne študije na različne
vrednosti oblikovnega parametra in števila vozlišč v lokalni poddomeni za model (I)
Poleg tega so simulacije izvedene na različno gostih razporeditvah vozlišč: 64x256,
128x512 in 192x786 z namenom demonstracije konvergence glede na gostoto
porazdelitve vozlišč. Študija vpliva Atwoodovega števila za vrednosti At = 0,01, 0,1,
0,3 in 0,5 na višino mehurčkov in intruzij je izvedena na modelih brez površinske
napetosti. Rezultati dobljeni na najfinejši mreži so primerjani z rešitvijo iz literature
dobljeno z zamaknjeno metodo označevalec-in-celica ter rešitvijo dobljeno z metodo
končnih volumnov s pomočjo odprtokodnega sistema za modeliranje toka tekočin
Gerris.
Brezmrežni rezultati kažejo, da je dinamiko Rayleigh-Taylorjeve nestabilnosti možno
učinkovito obravnavati s pomočjo kombinacije metode faznega polja in brezmrežnim
rešitvenim postopkom. S primerjavo rešitev vseh omenjenih modelov z referenčnimi
rezultati iz literature je pokazano, da površinska napetost pomembno spremeni obliko
premikajoče se meje med tekočinama ter, da se največja ukrivljenost pojavi na levem
in desnem repu gobam-podobnih nastalih struktur. Poleg tega občutljivostne študije
pokažejo, da so rezultati ob uporabi devetih vozlišč v lokalni okolici za vrednost
oblikovnega parametra c = 2,5 in 5 podobni, za primer c = 10 pa se medfazna meja
težje tekočine giblje hitreje kot medfazna meja lažje tekočine, poleg tega pa se pojavi
opazna ukrivljenost levega in desnega repa. Če za c = 10 uporabimo enajst ali trinajst
vozlišč v lokalni poddomeni rezultati ostanejo enaki ter se ujemajo z rezultati metode
končnih volumnov, kar kaže na to, da simulacije niso občutljive na število vozlišč v
lokalni poddomeni. Rezultati brezmrežne metode so tudi verificirani prek
reprodukcije dinamike premikajoče se meje, ki je konsistentna z referenčnimi
rezultati: za začetno simetrično perturbacijo meje se v primeru Boussinesqove
approksimacije simetrija ohrani še dolgo časa. V primeru modela z variabilno gostoto
pa je ta simetrija izgubljena kljub simetričnim začetnim pogojem.
Metoda je nato uporabljena za reševanje osnosimetričnega problema dveh so-
izlivajočih se, newtonskih, nestisljivih in ne mešajočih se tekočin z različnimi
fizikalnimi lastnostmi, pri čemer se pojavi ali kapljanje ali pa brizganje središčne
tekočine. Vrednotenje razvite metode je izvedeno na osnovi študije konvergence
izračunane brezdimenzijske dolžine curka glede na gostoto porazdelitve točk.
Brezmrežni rezultati so nato primerjani z rezultati dobljenimi z metodo končnih
volumnov kot je na voljo v odprtokodnem programu OpenFOAM. Občutljvostna
študija oblike curka na različne proste parametre kot sta razmerje viskoznosti ter
kapilarno število je bila prav tako izvedena in verificirana s primerjavo z
referenčnimi rezultati. Ugotovljeno je bilo, da v izračunani dolžini curka ni
pomembne razlike pri diskretizacijah: 30x400, 45x600 in 60x800 in rezultati z
vozliščno ureditvijo 30x400 so razumno točni. Brezmrežni rezultati prikazujejo
odlično ujemanje z rezultati dobljenimi z metodo končnih volumnov glede na
velikost kapljic ter časovni razvoj medfazne meje. Prav tako se rezultati ujemajo z
rezultati dobljenimi z metodo končnih razlik glede na brezdimenzijsko maksimalno
dolžino curka ter prostornino kapljic. Kombinacija metode difuznih približkov ter
metode faznega polja se izkaže za primerno za obravnavo vsiljenega
osnosimetričnega toka dveh tekočin.
Predstavljeno delo obravnava prvi poskus reševanja problema 2D Rayleigh-
Taylorjeve nestabilnosti in osnosimetričnega vsiljenega dvofaznega toka s
premikajočo se mejo z brezmrežno rešitvijo modela faznega polja. Rezultati pokažejo,
da je kombinacija metode difuzijskih približkov ter metode faznega polja sposobna
obravnavati tokovna polja z velikimi premiki in topološkimi spremembami medfazne
meje ter nudi učinkovito orodje za reševanje problemov hidrodinamike več
nemešajočih se tekočin. Poleg tega se je izvedlo več poskusov, da bi uporabili
metodo difuznih približkov z metodo faznega polja za modeliranje plinsko (s helijem)
fokusiranih mikro-curkov na podlagi plinskih dinamičnih virtualnih mikro-šob. Ti
poskusi so bili neuspešni, saj trenutni rešitveni postopek ne omogoča reševanja
tovrstnih problemov zaradi prevelikih razlik v gostotah med tekočino (vodo) in
plinom (helijem) ter mnogimi zelo ostrimi koti geometrije. Te ugotovitve kažejo na
potrebo po razvoju novih načinov za obravnavo ostrih kotov geometrije ter
izboljšanje uporabljene sheme za sklopitev hitrosti in tlaka.
Ključne besede
Dvofazni tok, proste in gibajoče se meje, računalniška dinamika fluidov, formulacija
faznega polja, dvodimenzionalni problemi, osnosimetrični problemi, brezmrežna
metoda difuznih približkov, Rayleigh-Taylorjeva nestabilnost, Boussinesqova
aproksimacija, spremenljiva gostota in viskoznost, fokusiranje toka, kapljanje,
brizganje
I
Table of Contents
List of Figures ..................................................................................................... V
List of Tables ..................................................................................................... IX
List of Symbols .................................................................................................. XI
Latin Symbols .................................................................................................... XI
Greek Symbols................................................................................................. XII
Acronyms and Abbreviations ....................................................................... XIII
1. Introduction ................................................................................................. 1
1.1 Free and Moving Boundary Problems .................................................. 1
1.2 Serial Femtosecond Crystallography and Flow Focusing ..................... 5
1.3 CFD Methodologies for Two-Phase Flow ............................................ 8
1.4 Literature Review of the Phase Field Method .................................... 12
1.5 Meshless Methods ............................................................................... 13
1.5.1 Diffuse Approximate Method ......................................................... 14
1.6 The Goals of the Dissertation.............................................................. 16
1.7 Overview of the Dissertation .............................................................. 16
2. Physical Model ........................................................................................... 19
2.1 Fluid Dynamics ................................................................................... 19
2.1.1 Single Phase Fluid Dynamics ......................................................... 19
2.1.2 Two-Phase Fluid Dynamics ............................................................ 23
2.2 Phase Field Model ............................................................................... 26
2.2.1 Sharp and Diffuse Interface ............................................................ 26
2.2.2 Phase Field and Free Energies ........................................................ 26
II
2.2.3 Cahn-Hilliard Dynamics .................................................................. 29
2.2.4 Surface Tension and Interface Width .............................................. 30
2.3 Hydrodynamic Coupling ..................................................................... 31
2.3.1 Phase-Fields Models for Large Density and Viscosity Ratios ........ 34
2.3.2 Boussinesq Approximation Model .................................................. 37
3. Numerical Method...................................................................................... 39
3.1 Characteristics of Meshless Methods .................................................. 39
3.1.1 Domain and Boundary Discretization ............................................. 40
3.2 Node Distribution and Local Subdomain ............................................ 42
3.3 The Approximation Function .............................................................. 42
3.3.1 The Collocation ............................................................................... 43
3.3.2 The Weighted Least Square Approximation ................................... 43
3.4 Spatial Discretization of Partial Differential Equations using Diffuse
Approximate Method ....................................................................................... 44
3.4.1 Construction of Local Interpolant using Polynomials ..................... 44
3.4.2 Calculation of Differential Operators .............................................. 47
3.4.3 Weight Function .............................................................................. 47
3.4.4 Upwind Scheme ............................................................................... 49
3.5 Time Discretization ............................................................................. 50
3.5.1 Explicit Euler Time Discretization .................................................. 50
3.6 Pressure-Velocity Coupling ................................................................. 51
3.7 Description of the Solution Procedure ................................................. 53
3.8 Numerical Implementation .................................................................. 56
4. Rayleigh-Taylor Instability Problem ........................................................ 57
4.1 Rayleigh-Taylor Instability Problem ................................................... 57
4.1.1 Problem Description and Literature Review ................................... 57
4.2 Governing Equations ........................................................................... 58
4.2.1 Problem Formulation ....................................................................... 58
4.2.2 Model Formulation .......................................................................... 59
III
4.2.3 Initial and Boundary Conditions ..................................................... 62
4.3 Results and Discussions ...................................................................... 62
4.3.1 Sensitivity Study with Respect to DAM Parameters ...................... 62
4.3.2 Effect of Atwood Number on the Height of Bubbles and Spikes ... 66
4.3.3 Comparison with Finite Volume Method ....................................... 71
5. Meshless Phase Field Method for Two-Phase Flow................................ 75
5.1 Governing Equations........................................................................... 75
5.1.1 Problem Formulation ...................................................................... 75
5.1.2 Model Formulation ......................................................................... 76
5.1.3 Initial and Boundary Conditions ..................................................... 78
5.2 Results and Discussions ...................................................................... 79
5.2.1 Sensitivity Study of Node Density .................................................. 79
5.2.2 Comparison with Finite Volume Results ........................................ 81
5.2.3 Sensitivity Study with Respect to the Process Parameters ............. 84
5.2.3.1 Effects of the Capillary Number .................................................... 84
5.2.3.2 Effects of the Viscosity Ratio ........................................................ 86
6. Conclusions ................................................................................................. 89
6.1 Summary of the Performed Work ....................................................... 89
6.2 Conclusions ......................................................................................... 90
6.3 Future Work ........................................................................................ 93
6.4 Publications ......................................................................................... 93
6.4.1 Journal Papers ................................................................................. 94
6.4.2 Conference Presentations ................................................................ 94
Appendix A Non-dimensional Form of the Governing Equations ......... 95
Bibliography ...................................................................................................... 97
Permissions to Reproduce Figures ................................................................ 117
Permission for Figure 1.3 .............................................................................. 117
IV
Permission for Figure 3.1 .............................................................................. 118
V
List of Figures
Fig. 1.1. A two fluid system involving moving boundary. .......................................... 2
Fig. 1.2. Classification of two-phase flow. .................................................................. 3
Fig. 1.3. Nanocrystals flow in their buffer solution in a gas-focused 4 μm diameter jet
at a velocity of 10 m/s perpendicular to the pulsed X-ray FEL beam that is focused on
the jet. (Chapman et al., 2011). (Reproduced with the permission of Nature
Publications)................................................................................................................. 6
Fig. 1.4. Schematic of GDVN tip, identifying its various parts and geometric
parameters. ,g lD D are the gas aperture diameter and inner diameter of liquid supply
capillary. H is the distance from the tip to the aperture and c is capillary tapering
angle (Beyerlein et al., 2015). (Reproduced from review of Scientific Instruments,
86, 125104, (2015); used in accordance with the Creative Common Attribution (CC
By) license). ................................................................................................................. 7
Fig. 2.1. Normal and tangential unit vectors on fluid/wall interface. ........................ 23
Fig. 2.2. Two-Phase fluid flow system separated by the interface 1,2 . .................... 24
Fig. 2.3. (a) Discontinuous physical properties across a sharp-interface. (b)
Continuous physical properties across diffuse interface. ........................................... 28
Fig. 2.4. Bulk free energy as a function of order parameter (see Eq. (2.24)). ...... 28
Fig. 2.5. Hyperbolic tangent profile for plane interface at an equilibrium. ............... 31
Fig. 2.6. Incompressible Navier-Stokes equations for two-phase flow together with
interface boundary conditions for sharp interface. ..................................................... 32
Fig. 3.1. Discretization of geometry for different numerical methods: (a) FEM, (b)
FVM, (c) FDM, (d) DRBEM and (e) MSM (reproduced with the permission of
Springer eBook publication). ..................................................................................... 41
VI
Fig. 3.2. Scheme of the discretization with the illustration of subdomains for the
boundary 1loc and the domain computational nodes
2loc . .................................... 42
Fig. 3.3.Scheme of central and upwind Gaussian weight function. The dots represent
the local subdomain. The blue curve is the original Gaussian center and the green
curve represents the upstream shifted Gaussian center. ............................................. 50
Fig. 3.4. Illustration of explicit time discretization scheme. ...................................... 51
Fig. 3.5. Block diagram of the solution procedure. ................................................ 55
Fig. 4.1. Scheme of the geometry, initial conditions and the boundary conditions of
the Rayleigh-Taylor instability problem. .................................................................... 59
Fig. 4.2. Model-I. Contours of RT instability for (left) 64x256 node arrangement and
(right) 128x512 node arrangement at 2.5, 5.0 and 10c for nine nodes in local
subdomain at 0.9 st . ............................................................................................... 65
Fig. 4.3. Model-I. Contours of RT instability for eleven (left) and thirteen (right)
nodes in local subdomain with 10,c for different node arrangements at 0.9 st .
.................................................................................................................................... 66
Fig. 4.4. Left: Initial phase field variable distribution in the cavity with 0.1moA
Right: The definition of height of the bubbles and the spikes. ................................... 67
Fig. 4.5. Time evolution of the interface for Model-II (a) At 0.1 , (b) At 0.3 , and
(c) At 0.5 using 11 nodes in local subdomain with 10c . .................................... 68
Fig. 4.6. Time evolution of the interfaces for Model-III, At 0.5 using 11 nodes in
local subdomain with 10c . .................................................................................... 69
Fig. 4.7. Left: Model-II. Right: Model-III. Interfaces for At 0.5 at 1.1s.t Both
simulations are done with 11 nodes in a subdomain and 10c . ............................... 70
Fig. 4.8. Left: The height of the bubbles hb versus the height of the spikes hs for
At 0.01, 0.1, 0.3 and At 0.5 of Model-II and Model-III. The solid lines represent
the results (Lee and Kim, 2012) and markers show the present results. Right: A
comparison of inter-fluid boundary of Model-II and Model-III for At 0.5 at
1.1s.t ...................................................................................................................... 71
VII
Fig. 4.9. A comparison of DAM (Model-I) and FVM results at different times using
128 x 512 node arrangement. The solid and dashed lines represent FVM and DAM
results with 11 points in local subdomain and 10c , respectively. ......................... 73
Fig. 4.10. Model-I. Time evolution of the moving boundary by using 64 x 256 node
arrangement with 11 nodes in local subdomain for 10c . ..................................... 74
Fig. 4.11. Model-I. Time evolution of the interface by using 128 x 512 node
arrangement with 11 nodes in local subdomain for 10c . ..................................... 74
Fig. 5.1. Diagram scheme of the geometry. ............................................................... 76
Fig. 5.2. Illustration of the definition of the jet length L j and the limiting length Ld.
.................................................................................................................................... 80
Fig. 5.3. Dimensionless jet length L j as a function of dimensionless time t for the
different node arrangements....................................................................................... 81
Fig. 5.4. A comparison of DAM and FVM results using 0.44 m / s, 0.3 m / si ov v
at (a) 1 mst and (b) 2 ms.t ................................................................................ 83
Fig. 5.5. A comparison of DAM and FVM results using 0.44 m / s, 0.9 m / si ov v
at (a) 1 mst and (b) 2 ms.t ................................................................................ 83
Fig. 5.6. Variation of limiting length Ld (left) and volume of the drop Vd (right) as a
function of capillary number. The solid lines represent the finite difference results
(Liu and Wang, 2015) and the markers show the results from this study. ................. 85
Fig 5.7. Dimensionless limiting length Ld as a function of dimensionless time t for
(a) Ca 0.01 and (b) Ca 0.05 for 0.1 . ......................................................... 85
Fig. 5.8. The interface profile for (a) Ca 0.01 at 87t and (b) Ca 0.05 at
74t for 0.1 . ................................................................................................. 86
Fig. 5.9. Variation of limiting length Ld (left) and volume of the drop Vd (right) as a
function of viscosity ratio. The solid lines represent the finite difference results (Liu
and Wang, 2015) and the markers show the present results. ..................................... 87
VIII
Fig. 5.10. Dimensionless limiting length Ld as a function of dimensionless time t
for (a) 1.0 and (b) 2.0 at Re 100 . ......................................................... 87
Fig. 5.11. The interface profile for (a) 1.0 at 82.9t and (b) 2.0 at
106.9t for Re 100 . ............................................................................................. 88
IX
List of Tables
Table 4.1. Material Properties used in simulations with Model-I. ............................ 63
Table 4.2. Material properties used in simulations with Model-II and Model-III. ... 67
Table 5.1. Material properties used in simulations. .................................................. 80
X
XI
List of Symbols
Latin Symbols
v velocity vector
p position vector
P Pressure
t Time
, ,x y zi i i unit vectors in x, y, z directions
, ,x y zp p p Cartesian coordinates
V Volume
T stress tensor
sf surface force
bf body force
gf gravitational force
g gravitational acceleration
Pf pressure force
f viscous friction force
n̂ unit normal vector
t̂ unit tangential vector
fv velocity of fluid
wallv velocity of wall
1 2,v v velocities of fluids 1 and 2
nv normal component of velocity
XII
mixF mixing energy
0f bulk energy density function
bulkF bulk distortion energy
1 2 3, ,K K K elastic constants for splay, twist and bend
anchF anchoring energy
F total free energy
j Interfacial diffusive flux
M Mobility
stf surface tension force
buof buoyancy force
nx central node of local subdomain
J number of nodes in local subdomain
bN number of basis functions
l N number of nodes in local subdomain
*v intermediate velocity
L j dimensionless jet length
Vd dimensionless volume of drop
Ld dimensionless limiting length
Greek Symbols
Density
τ viscous friction
τ deviatoric stress tensor
dynamic viscosity
bulk or volume viscosity
s fixed wall
XIII
surface tension
mean curvature
phase field variable
, phenomenological parameters in free energy
interface width
mixing energy density
regularization parameter
boundary of the domain
computational domain
chemical potential
1 2, densities of both fluids
1 2, viscosities of both fluids
* constant density
basis functions
set of basis function
weight function
Acronyms and Abbreviations
CFD Computational Fluid Dynamics
NSE Navier-Stokes Equation
MAC Marker-And-Cell
SMMC Surface Marker-And-Micro-Cell
VOF Volume Of Fluid
SLIC Simple Line Interface Calculation
PLIC Piecewise Linear Interface Construction
FLAIR Flux Line-segment Advection and Interface Reconstruction
FCT Flux Correct Transport
SL-VOF Segment Lagrangian-Volume of Fluid
XIV
LSM Level Set Method
ALE Arbitrary Lagrangian-Eulerian
FT Front Tracking
PFM Phase Field Method
DI Diffuse Interface
PDE Partial Differential Equation
FDM Finite Difference Method
FEM Finite Element Method
FVM Finite Volume Method
BEM Boundary Element Method
MM Meshless Method
EFGM Element Free Galerkin Method
MLPGM Meshless Local Petrov-Galerkin Method
SPIM Smoothed Point Interpolation Method
MLRPIM Meshfree Local Radial Point Interpolation Method (RPIM)
RBFCM Radial Basis Function Collocation Method
LRBFCM Local Radial Basis Function Collocation Method
MFS Method of Fundamental Solutions
DAM Diffuse Approximate Method
2D Two-dimensional
NSCH Navier-Stokes- Cahn-Hilliard
NSK Navier-Stokes-Korteweg
1
Equation Chapter (Next) Section 1
1. Introduction
1.1 Free and Moving Boundary Problems
Multiphase flow is simultaneous flow of matter with different phases such as gas,
liquid and solid or with different non-mixing substances with the same phase, i.e.,
liquid-liquid like oil-water. Multiphase flow is quite common phenomena that occurs
both in nature and in technology. The most trivial example in nature is that of the
clouds, where the droplets of liquid are moving in the gas. Furthermore, the melting
of the polar ice caps and manufacturing of nano-materials are two typical examples.
Multiphase flows play an important role in industrial applications such as energy
conversion, paper manufacturing and food manufacturing. They also occur in various
environmental phenomena like rain, fog, snow, soil erosion, landslides and
biomedical flows like blood flow. Multiphase flow also plays an important role in
nuclear power plants, combustion engines, propulsion systems, chemical industry, oil
and gas production. Multiphase flows are also encountered in different types of
equipment such as furnaces, distillation and bubble column, stirred vessels and
engine injection and coolant systems. Multiphase flows give rise to very complex
combinations of phases as well as flow structures. A simplest case of multiphase
flow is the two-phase flow and in many cases the proper functioning of most of the
equipment crucially depends on the existence of two-phase flow. Therefore, the
understanding and analysis of two-phase system is of great importance if processes
involving two-phase system need to be safely designed and controlled. A large
number of two-phase flow problems in science and engineering are formulated in
terms of time dependent Partial Differential Equations (PDEs) with moving
boundaries or interfaces. In the mathematical model, there is a presence of initially
2
unknown free boundary or a boundary which moves throughout the analysis, the
determination of which is an important part of the solution procedure. Usually, the
term “free boundary” is used when the boundary is stationary and steady-state
problem exists. Moving boundary, on the other hand, is used for time dependent
problems and position of boundary has to be determined as a function of time and
space. Two-phase flow problems having free and moving boundary problems are
challenging because of the complexity associated with the deformed interfaces or
broken surfaces, the multiple time and length scale and non-linearity associated with
coupling of the dynamics of the interface with the dynamics of the material (Li,
2006). A schematic diagram of two fluid system involving moving boundaries and
free surface is shown in Fig. 1.1. Practical examples of two-phase flow having free
and moving boundary problems are piston driven flows, fluid-fluid interface, wetting,
capillary flows, bubble and droplet deformation and oscillation, glass forming and
coating of solid substrates, binary alloy solidification and melting, recrystallization of
metals, epitaxial growth of thin film and nozzle problems (Brennen, 2005).
Fig. 1.1. A two fluid system involving moving boundary.
The term two-phase covers a wide range of flow patterns and regimes in engineering
and chemical process, and are categorized by the physical states of the constituent
components present in the system and by the topology of the interfaces. Thus, two-
phase flows can be gas-solid, gas-liquid, and solid-liquid (see Fig. 1.2) or in case of
two immiscible liquids, liquid-liquid. Similarly, the two-phase flow is also classified
topologically as dispersed, transitional and separated (Rusche, 2003). This variety of
3
combinations makes an extremely difficult task for the design of industrial equipment
for two-phase applications. Flow of mud, flow of liquid with suspended solids such
as slurries, fluid motion in aquifers are examples of solid-liquid flow. The main idea
of two-phase flow system is that one phase acts as the continuous phase and other is
dispersed phase.
Fig. 1.2. Classification of two-phase flow.
The production of oil from the ground, the coalescence of Newtonian and non-
Newtonian drops in shear flows, spreading of liquid drops on solid surfaces with
three phase contact line and the steady flow of viscoelastic film over a periodic
topography under the action of body force (Vasilopoulus, 2016) are the examples of
two-phase flow. Immiscible liquid-liquid flow has many industrial applications such
as liquid extraction processes and dispersive flows. In dispersive flows, liquids can
be dispersed into droplets by injecting a liquid through an orifice or a nozzle into
another continuous liquid. The injected liquid can start dripping or can forms a long
steady jet at the outlet of the nozzle depending upon the flow rate of injected liquid
and continuous liquid. For small flow rate liquid may drip continuously and for high
flow rate liquid forms a continuous jet at the exit of the nozzle. In other applications,
the injected liquid can be dispersed as tiny droplets into another liquid to make an
emulsion. Emulsions are of great importance in a variety of applications such as food,
4
chemistry, pharmaceutics, industry and environmental science (Wu et al., 2017). For
the analysis of chemical and biological samples, the manipulation of monodispersed
emulsion has become crucial (Anna et al., 2003). Among all the possible approaches,
the axisymmetric microfluidics system (Gañán-Calvo and Gordillo, 2001; Herrada et
al., 2010) gives promising ways for producing highly monodispersed emulsions. It is
important to understand comprehensively the dynamic behaviors of drop formation
in axisymmetric microfluidics to optimally design the microfluidic system and
precisely manipulate the droplet production. The typical axisymmetric microfluidics
are co-flow microfluidic device (Utada et al., 2007) and flow-focusing microfluidic
device (Gañán-Calvo and Gordillo, 2001), where both continuous phase and
dispersed phase are coaxial. The main advantage of axisymmetric devices is that
there is no wetting problem between two phases, which can effect the droplets (Wu et
al., 2017).
The generation of liquid jets with diameters of micron or sub-micron is of high
relevance for various applications in industry, medicine and technology such as
microfiber spinning, inkjet printing, micro-analytical dosing of liquids and mostly for
pharmaceutical and microbioanalytics (Trebbin et al., 2014). They are produced by
focusing of a fluid by another co-flowing immiscible lower viscosity fluid (Eggers,
1997; Eggers and Villermaux, 2008). Microjets are a natural antecedent of the drops,
bubbles, emulsions and capsules used in various technological applications (Basaran,
2002). Microjet production techniques must satisfy the existence of a robust steady
jetting regime, stable over a wide range of experimental conditions and control of jet
features through operational parameters. The flow focusing technique (Gañán-Calvo,
1998) in formation of jetting mode uses the pressure gradient exerted by the outer gas
stream to focus a steady liquid meniscus whose tip emitted a microjet. The jetting-
dripping transition of a flow focused viscous liquid jet surrounded by the co-flowing
immiscible lower viscosity liquid at minimum flow rate were studied in (Gañán-
Calvo, 2006). The stability of flow focusing takes place in converging-diverging
nozzle. The size of the microjets was experimentally examined in (Acero et al., 2012).
A theoretical and experimental research to investigate both the atomization dynamics
of non-Newtonian liquids as well as the performance of coaxial atomizers utilized in
pharmaceutical tablet coating was analyzed in (Aliseda et al., 2008). The cone-jet
patterns and the transition from jetting to dripping was analyzed in (Herrada et al.,
5
2008). Eggers and Villermaux (Eggers and Villermaux, 2008) have made an
extensive literature review in the field of liquid jet behavior, describing the physical
phenomena of its breakup. The liquid jet breaks into drops due to the growth of
axisymmetric instabilities, namely, the Rayleigh-Plateau instability (Rayleigh, 1878).
The most recent microfluidics application where long, thin and stable micro-jets are
required is Serial Femtosecond Crystallography (SFX) (Chapman et al., 2011).
1.2 Serial Femtosecond Crystallography and Flow
Focusing
In SFX, highly coherent femtosecond X-ray pulses created by X-ray Free Electron
Laser (XFEL) scatter of the protein microcrystals transported into the X-ray beam via
a micron thin liquid jet. The X-ray diffraction patterns, collected before crystal
destruction, are used to obtain their internal molecular structure (Chapman et al.,
2011). However, it is confirmed that the sufficiently short femtosecond pulses able to
collect data before the onset of substantial damage, called “diffraction-before-
destruction” (Chapman et al., 2011). SFX experiment setup shown in (Fig. 1.3)
requires a controlled delivery of samples by steady continuous jets or monodispersed
stream of liquid droplets. These protein crystal samples are sensitive to solvation
conditions and difficult to crystallize, so they are sustained in their native
environment, special buffer solution, for possibly before they are inserted into the
vacuum, where they are illuminated by an X-ray beam (Beyerlein et al., 2015). It is
critical to keep the surrounding background signal emitting from the buffer solution
as low as possible because of the weak scattering ability of samples. So, the jet
diameters should be as thin as possible, comparable to the diameter of X-ray beam
(~1.0 µm). It is required for SFX experiments that the samples move faster through
the X-ray beam to avoid the double exposure, operating with a repetition rate of
120Hz to few KHz frequency. Hence, for the best possible results of SFX
experiments the sample carrier jet has to be thin, long and fast.
6
Fig. 1.3. Nanocrystals flow in their buffer solution in a gas-focused 4 μm diameter jet
at a velocity of 10 m/s perpendicular to the pulsed X-ray FEL beam that is focused on
the jet. (Chapman et al., 2011). (Reproduced with the permission of Nature
Publications).
As micro-jets are preferable for the delivery of protein samples in SFX experiments
so Rayleigh jet nozzles proved to be ineffective to produce jets with smaller diameter
due to the strong correlation between the jet diameter and nozzle outlet diameter. In
addition, they are only capable of producing jet or drop diameters of ~20 µm or
above and are prone to clogging to deliver jets or drops with smaller diameter
(Weierstall, 2014). The Gas Dynamic Virtual Nozzle (GDVN) produces much
smaller jets as compared to the conventional plate-orifice apparatus. In SFX
experiments, GDVN (see Fig. 1.4) has been used to produce micrometer-sized
streams using the focusing action of coaxial sheath gas (Beyerlein et al., 2015). Very
recently, the same GDVN has been used for the numerical simulation of gas focused
liquid jets (Zahoor, 2018). A numerical study has also been carried for investigating
the effects of nozzle geometry on stability, shape and flow characteristics of micron-
sized liquid jets produced by GDVN (Zahoor et al., 2018). Furthermore, in order to
provide efficient and reliable delivery of fresh crystals across the beam of XFEL in
SFX experiment, an experimental investigation and numerical simulation has been
performed using double flow focusing nozzle (Oberthuer et al., 2017).
7
Fig. 1.4. Schematic of GDVN tip, identifying its various parts and geometric
parameters. ,g lD D are the gas aperture diameter and inner diameter of liquid supply
capillary. H is the distance from the tip to the aperture and c is capillary tapering
angle (Beyerlein et al., 2015). (Reproduced from review of Scientific Instruments,
86, 125104, (2015); used in accordance with the Creative Common Attribution (CC
By) license).
Generally, it is difficult to acquire experimental data from the existing industrial
processes as they are often carried out at elevated pressure and temperature or might
employ hazardous substances. In addition, the disturbance caused by the installation
of measuring devices is often unacceptable. Hence, the design processes, thirty years
ago, mainly rely on experimental pilot scale studies and empirical correlations
(Bergles et al., 1981). Pilot scale studies are performed on a smaller scale and
regularly at ambient pressure and temperature, and also utilizing convenient
modelling fluids. The experiments are usually time consuming and expensive. It is
also required for the pilot scale study to use the scaling laws to the full-size plant,
which may not be well settled (Bergles et al., 1981). On the other hand, the main
8
disadvantage of empirical correlations is that the experimental information is
encoded in global parametric form, which effectively disguises the detailed localized
information. For the applications of two-phase flow, these difficulties are very
noticeable due the increased number of fluid properties as well the variety of flow
regimes and patterns.
From the above discussion, it is obvious that the development of the methodology
that predicts the entire field of flow with sufficient accuracy is highly desirable,
which exists in the form of Computational Fluid Dynamics (CFD). The numerical
simulation using CFD has emerged a powerful numerical tool for understanding the
dynamics of two-phase system. CFD is the analysis of engineering systems including
fluid flow, heat transfer and associated phenomena by means of computer-based
simulation.
1.3 CFD Methodologies for Two-Phase Flow
A numerical methodology consists of physical model and solution procedure. A
physical model is the mathematical representation of the set of governing equations,
involving physical or/and chemical process to be simulated or predicted. Usually,
less influential or less important phenomena is neglected in models. The solution
procedure identifies the details about finding the approximate solution from the
model equations numerically. Traditionally, the dynamics of two-phase flows
experienced in engineering processes are modelled by the Navier-Stokes Equations
(NSEs) augmented by the Newtonian law of viscosity and an equation of state for
density and pressure.
The computational modelling of two-phase problems has become a highly popular
research subject due to its pronounced influence in improving our better
understanding of the nature and the development of the advanced technologies. An
important feature in two-phase flows is the existence of deformed interfaces or
boundaries that separate both phases. The topology of the interface constantly
changes as the phases interact with each other exchanging mass, momentum and
energy. The main difficulty is to handle the complexity of interface topology and the
fact that the interfacial location is a priori unknown. Consequently, the detailed
9
description of moving boundaries or interfaces is a challenging task from physical as
well as computational point of view due to the strong effect of surface tension and
discontinuities that arise in the stress and pressure fields across the interface.
According to (Fuster et al., 2009), two-phase flow problems for practical applications
exhibit all or several of the following characteristics: high surface tension, low
viscosity, high density ratios, complex and evolving interface topologies and spatial
scale ranging over several order of magnitude. Therefore, an ideal numerical method
in order to solve NSEs together with surface tension effect would have the following
properties:
robust representation of evolving, topologically complex interfaces,
accurate representation of surface tension, which requires accurate normal
and curvature of interface,
robust and accurate handling of large density and viscosity ratios,
efficient representation of evolving flow features of widely different
characteristics spatial scales.
In recent years, advanced numerical techniques have been developed to simulate the
two-phase flows. One main difference between these methods is the representation of
interface and calculation of curvature and normal at the interface. All methods can be
divided into two classes for the representation of the interface. In the first class, the
interface is represented implicitly by a function defined on all of the domain and in
the second class the interface is explicitly tracked. Numerical methods for the
simulation of multiphase flows, are classified based on the type of flow modelling
(Eulerian, Lagrangian, mixed), the type of interface modelling (capturing or tracking),
and the type of spatial discretization (finite difference, finite volume, finite element,
boundary element, meshless and others).
A conceptually straightforward way to handle the moving boundaries is to employ a
moving mesh that has the grid points on the interface and deforms as determined by
the flow on both sides of the boundary. This has already been implemented in
Boundary Element Method (BEM) (Khayat, 2000; Toose et al., 1995), Finite
Element Method (FEM) (Hu et al., 2001) and Finite Difference Method (FDM)
(Ramaswamy and Leal, 1999) . The main drawback of these methods is that they
cannot easily handle the singular topological changes of interface such as breakup or
10
coalescence and mesh entanglement caused by the large displacement of internal
domains. As an alternative, fixed grid (Eulerian) methods have been developed such
as Marker-And-Cell (MAC) method (Harlow and Welch, 1965), where Lagrangian
massless markers are used to identify each fluid phase. It is an approach to decouple
pressure and velocity in solving NSEs for time dependent incompressible flow with a
free surface. The shape and location of the free surface is determined by tracking the
movement of these markers, which are advected with the local fluid field. An
improved version of this method, namely, Surface Marker-and- Micro-Cell (SMMC)
was introduced by (Chen et al., 1997). In this method, markers are placed only along
the interface in order to reduce the total arithmetic of tracking markers throughout the
flow domain. The MAC methods are easy to implement but have great limitations for
severe topological changes of interface such as interface merging, breaking up and
overturning.
Afterwards, the invention of Volume of Fluid (VOF) method (Hirt and Nichols,
1981) represented a milestone in the simulation of multiphase flow. In this method, a
scalar volume fraction is introduced and its value varies from zero to one. One
represents a full cell and for empty cell zero value is used. The values between zero
and one represents the interface. In VOF method, an interface needs to be
reconstructed based on discrete values of volume fraction and then advanced with
local velocity field to a new time step. Originally, Donor and Acceptor method,
which is a Simple Line Interface Calculation (SLIC) method, was used for interface
reconstruction with lower accuracy (Noh and Woodward, 1976). For the better
representation of interface, new methods with higher accuracy have been developed
such as Piecewise Linear Interface Construction (PLIC) scheme (Youngs, 1982) ,
Flux Line-segment Advection and Interface Reconstruction (FLAIR) (Ashgriz and
Poo, 1991), Flux Correct Transport (FCT) (Rudman, 1997) and Segment Lagrangian-
Volume of Fluid (SL-VOF) (Guignard et al., 2001). An important feature of VOF
method is that it conserves mass precisely as it is a conservative method. The main
disadvantage of VOF is the reconstruction of interface after each time step, which is
computationally demanding task.
Based on the similar concept as VOF, (Osher and Sethian, 1988) introduced simple
and versatile Level Set Method (LSM) for multiphase simulations, where an interface
is defined by a level set function, initialized as a signed distance function from the
interface, positive on one side and negative on other side of the interface. The
11
interface is represented by the zero-level of the level set function. In this method, the
computation of curvature and surface tension is straightforward due to smoothing
characteristics across the interface. The main drawback of LSM is that the fronts
evolve as a solution of transport equation for level set function, which causes the
level set function to lose the distance function properties at later times based on the
calculated velocity fields. This appears a smearing of the interface and causes
difficulty to ensure mass conservation. In order to overcome this difficulty, some
mass correction schemes have been proposed in (Zhang et al., 2010), which increases
the computational cost. VOF (Scardovelli and Zaleski, 1999) and LSM (Sussman et
al., 1994) method using implicit representation of the interfaces can efficiently and
robustly handle the evolving, topological complex interfaces but generally suffer
from accurate representation of surface tension (Popinet and Zaleski, 1999). On the
other hand, the methods using explicit representation of interfaces such as Arbitrary
Lagrangian-Eulerian (ALE) (Fyfe et al., 1988) and Front Tracking (FT) (Shin et al.,
2005) can provide accurate representation of surface tension but have difficulty while
tackling the complex, evolving interface topologies.
Apart from these aforementioned methods, there is another Eulerian approach to
handle the complex topological interface between two phases is Phase Field Method
(PFM) (Anderson et al., 1998). PFM differs from the aforementioned methods by
assuming that the interface is diffuse in a physical rather than numerical sense. It
provides a useful tool for capturing the evolution of complex interfaces and treating
the topological changes of the interface. In the PFM, an interface is described as a
finite volumetric zone across which the physical properties (density, viscosity, phase
field variable, etc.) vary steeply and continuously. The shape of the interface is
determined by minimizing the free energy of the system (Cahn and Hilliard, 1958),
no explicit interphase boundary condition is required at the moving boundary. The
surface tension appears as a surface free energy per unit area caused by the gradient
of the phase field variable. In addition, PFM does not require any conventional
algorithms like Donor-Acceptor (Hirt and Nichols, 1981), Flux Line-segment
Advection and Interface Reconstruction (FLAIR) (Ashgriz and Poo, 1991), for
reconstruction and advection of an interface. It constructs the interface by taking the
gradient of the chemical potential into account, so the effect of surface tension on
12
flow fields can be treated without complicated topological calculations of the
interfacial profile.
1.4 Literature Review of the Phase Field Method
In this dissertation, PFM is used in order to analyze the dynamics of two-phase flows.
PFM was discovered more than two decades ago to deal with the difficult problem of
crystal growth. The phase-field approach has been extended to the dendritic growth
(Kobayashi, 1993) and stress-induced instabilities in solids (Kassner and Misbah,
1999) from the original work of (Collins and Levine, 1985) for the simulation of
diffusion-limited crystal growth. It has also been used for the mesoscale simulation
of the solidification of a binary alloy (Bi and Sekerka, 1998), polymer membrane
formation in a highly functional material design platform project (Morita et al., 2001).
PFM has also ability to efficiently simulate the complex two-phase flows than other
methods, as the computational cost of PFM simulations does not depend on the
interfacial deformation, but on the spatial and temporal resolutions. PFM has been
also successfully implemented in fluid mechanics problem such as Marangoni
convection (Borcia and Bestehorn, 2003), droplet and vesicle dynamics (Beaucourt et
al., 2004) and polymer blends (Roths et al., 2002).
Besides solidification (Boettinger et al., 2002) and phase transformation (Chen,
2002), PFM has also been used for simulating grain growth and phase separation
(Gomez et al., 2010), crack propagation (Henry and Levine, 2004). Later on, it is also
used to model foams, planet formation, ferroelectric ceramics, growth of cancerous
tumors, dewetting and rupture of thin liquid films, solid-solid transitions, phase
separation of block copolymers, infiltration of water into a porous medium and
dendritic growth (Gomez and Hughes, 2011). Recently, it has gained much
popularity for analyzing the dynamics of two-phase fluid flow problems (Anderson et
al., 1998; Liu and Shen, 2003) and has been used for wide range of two-phase
problem such as Hele-Shaw flow, moving contact lines, head-on droplet collision
(Yue et al., 2004), capturing of local two-phase interface (He and Kasagi, 2008),
pinchoff of liquid-liquid jets (Kim et al., 2009), spreading of micro-sized droplet on
heterogeneous surface (Lim and Lam, 2014), dripping-jetting transition depending on
the Capillary number of outer fluid and Weber number of inner fluid (Liu and Wang,
13
2015). A phase field models has also been developed by using the Boussinesq
approach to simulate the three-phase flows (Kim and Lowengrub, 2005) and a semi
discrete Fourier-spectral method has been used to approximate the phase field model
based on the Boussinesq approximation for the mixture of two immiscible fluids
(Liu and Shen, 2003). The phase field model with different density and viscosity of
the phases has been used to simulate the incompressible two phase system (Dong and
Shen, 2012). A detailed review for the development of PFM can be found in (Kim,
2012) . Recently, a PFM has been developed for compressible binary mixtures based
on the balance of mass, momentum, energy and second law of thermodynamics. It
has been proved analytically and numerically that the developed model is capable to
describe the phase equilibrium for binary mixture of CO2 and ethanol by changing
the parameters of the model, which measures the attraction force between molecules
of both components (Liu et al., 2016). Furthermore, a thermal phase field model
based on Navier-Stokes-Korteweg (NSK) equation has been developed and solved to
analyze the phase transitions of a droplet and thermocapillary convection (Park et al.,
2018).
1.5 Meshless Methods
Two-phase flows are prescribed by PDEs, which are highly non-linear, time
dependent and fully coupled, and is difficult to solve analytically. So, a wide variety
of mathematical theories and computational technologies have been developed for
the accurate and efficient numerical solutions. Traditional numerical methods such as
FDM (Ozisik, 1994), FEM (Zienkicwicz and Taylor, 2000), Finite Volume Method
(FVM) (Versteeg and Malalasekera, 2007) and BEM (Hall, 1994) are well
established and used to solve physical models in science and engineering. Despite the
powerful features of these methods, there are often substantial difficulties in applying
them to realistic, geometrically complex three-dimensional transient problems. A
common drawback of FDM is that the discretization needs to align with the
coordinate lines whereas in case of FEM and FVM one needs to create a
polygonisation, either in the domain and/or on its boundary. This type of meshing is
often the most time consuming part of the solution procedure and is far from being
fully automated. In order to avoid the problem of polygonisation, a number of
14
meshless methods have been developed in the recent years (Atluri and Shen, 2002; Li
and Mulay, 2013).
There exist several Meshless Methods (MMs) but the common ones are Element Free
Galerkin Methods (EFGM) (Belytschko, Lu, et al., 1994), the Meshless Local
Petrov-Galerkin Method (MLPGM) (Atluri and Shen, 2002), Smoothed Point
Interpolation Method (SPIM) (Liu, 2002), Meshfree Local Radial Point Interpolation
Method (MLRPIM) (Liu et al., 2002), the smoothed particle hydrodynamics (Liu and
Liu, 2003), the Radial Basis Function Collocation Method (RBFCM) (Kansa, 1990a,
1990b), Local Radial Basis Function Collocation Method (LRBFCM) (Šarler and
Vertnik, 2006), Method of Fundamental Solution (MFS) (Chen et al., 2008) and
Diffuse Approximate Method (DAM) (Nayroles et al., 1991).
1.5.1 Diffuse Approximate Method
In this dissertation, the emphasis is on the application of DAM, which uses Weighted
Least Squares (WLS) to determine locally smooth and differentiable approximation
of discrete data. This method has been first proposed by (Nayroles et al., 1991) and
afterward generalized in (Belytschko, Lu, et al., 1994) and (Belytschko et al., 1996)
and named it Element Free Galerkin (EFG) method. Fracture crack growth problem
was calculated in (Belytschko, Gu, et al., 1994), where different continuous and
discontinuous weight functions were examined to determine the influence on the
simulation of a crack. Afterwards, this method was further developed by the group of
Professor Hamou Sadat for various applications such as natural convection in porous
media (Prax et al., 1996), solution of Navier-Stokes equations (Couturier and Sadat,
1998), solid/liquid phase change phenomena (Bertrand et al., 1999), lid driven cavity
benchmark (Sadat and Prax, 1996) and solution of radiative transfer equation with
discrete ordinates approach (Sadat, 2006).
The group of Professor Hamou Sadat has also successfully carried out a comparative
study of DAM and control-volume based finite element method (Prax et al., 1998),
natural convection in fluids (Couturier and Sadat, 1998a) and the performance and
accuracy of DAM for 2D natural laminar convection in fluids (Sadat and Couturier,
2000). The three dimensional natural laminar convection was carried out in (Sophy et
al., 2002) and heterogeneous heat conduction problems in (Sadat et al., 2006). The
15
solution of coupled radiative and conductive heat transfer in complex multi-
dimensional geometries using DAM was found in (Sadat et al., 2012). DAM has also
been implemented for three-dimensional fluid flow and heat transfer problems in
(Wang et al., 2012). The vorticity and vector potential formulation of NSEs has been
used to avoid the difficulties of pressure velocity coupling. The implicit time
integration, Gaussian weight function and second order polynomial with number of
nodes in local subdomain between 27 and 40 have been used.
Possibly, the first industrial application of DAM was elaborated in (Šarler et al., 2004)
for calculation of solid-liquid phase change phenomena in direct chill casting of
aluminum slabs. Furthermore, it has been used with second order polynomial,
Gaussian weight function and nine-noded local subdomains for the steady state
convective-diffusive solid-liquid phase change problem associated with temperature
fields in direct-chill, semi-continuously cast billet and slabs from aluminum alloys
(Šarler et al., 2005). It has also been employed for modelling of transport phenomena
in porous media (Perko, 2005), calculation of radionuclide transport (Perko and
Šarler, 2005). The time-dependent Burgers equation has been using DAM on non-
uniform computational node arrangement. The Gaussian weight function and second
order polynomial with nine-noded local subdomains. It was proved that the stability
of the solution depends on the shape parameter of the weight function and
randomness of the node arrangement (Perko and Šarler, 2007). Recently, it is shown
that DAM can be used to solve the NSEs in complex-shaped computational domain
by analyzing the dynamics of lid driven cavity and backward facing step problems on
non-uniformly distributed computational nodes (Kosec, 2016). DAM has also been
used for the simulations of low frequency electromagnetic casting (Hatić, Mavrič,
Košnik, et al., 2018; Košnik et al., 2016) with second order polynomial, Gaussian
weight function and thirteen-noded local subdomains. Furthermore, the DAM has
been applied for the simulation of macrosegregation in a solidifying cavity (Hatić,
Mavrič, and Šarler, 2018). Where, the explicit time stepping, Gaussian weight
function, second order polynomial with thirteen-noded subdomains are used. The
pressure-velocity coupling is performed using fractional step method (Chorin, 1968).
The upwind approach, for the first time, has been introduced with the shift of
Gaussian function in the opposite direction of velocity field, which is also employed
in the present work.
16
1.6 The Goals of the Dissertation
The main goal of the present dissertation is the development of meshless DAM in
connection with PFM to solve coupled Navier-Stokes Cahn-Hilliard equations, which
govern the dynamics of moving boundary problems. The objectives of the performed
research work are
Development of the related numerical model for solving 2D Rayleigh-Taylor
instability problem for three different physical models: a model with large
density variation and surface tension, Boussinesq formulation for small
density variation, and phase field dependent density for small density
variation without the surface tension. The sensitivity analysis of DAM
parameters in terms of shape parameter values, number of nodes in local
subdomain and node density convergence. To study the effect of surface
tension and dimensionless Atwood number on the dynamics of Rayleigh-
Taylor instability.
Development of the numerical model in axisymmetry to analyse the dripping
and jetting phenomena. To study the effect of node arrangement on
dimensionless jet length. To analyse the effect of flow rates of inner and outer
fluid for dripping and jetting phenomena. To study the effect of capillary
number and viscosity ratio on the dimensionless limiting length and volume
of drop.
To validate the numerical model by the numerical results obtained by other
numerical methods.
1.7 Overview of the Dissertation
In Chap. 2. , the physical phenomena for single and two-phase flow is
described. The mass and momentum conservations equations for single phase
flow together with appropriate boundary conditions are presented. For two-
17
phase sharp interface, the governing equations and interface boundary
conditions are discussed. The detailed description of PFM based on free
energy for two-phase flow is elaborated, in terms of surface tension effect in
interfacial region and hydrodynamic coupling of momentum and CH
equation.
The meshless numerical method DAM is presented in Chap. 3. by discussing
the construction of local interpolant with polynomials and calculation of
differential operators. The Gaussian weight function, shape parameter and
upwind approach is also discussed. The explicit Euler time discretization
scheme and pressure velocity coupling algorithm is presented.
In Chap. 4. , the numerical method is verified for 2D benchmark Rayleigh-
Taylor instability problem using three different physical models. The
sensitivity study of shape parameter, number of nodes in local subdomain and
node density convergence is elaborated. It also includes the comparison of
meshless results with FVM results.
Phase field simulation of two co-flowing immiscible fluids in axisymmetry
with different material properties that yield dripping or jetting of the core
fluid is presented in Chap. 5. The node density convergence in terms of
calculated jet length is presented. Dripping and jetting phenomena are
analyzed and compared with FVM results.
Finally, conclusions and further developments are presented in Chap. 6.
Equation Chapter (Next) Section 1
18
19
2. Physical Model
In this chapter, the fundamental principles and governing equations for single and
two-phase flow are explained. The basic governing equations and boundary
conditions for describing the motion of single fluid is presented. The governing
equations for sharp fluid/fluid interface together with appropriate interface boundary
conditions are described. An overview of existing theories that describe the nature of
interface between two phases is presented. Furthermore, the PFM based on free
energies is presented for diffuse interface. A phase field model for large density and
viscosity ratios is also presented. At the end the Boussinesq phase field model is
discussed.
2.1 Fluid Dynamics
The main governing equations for fluid dynamics are the classical NSEs developed
by Claude-Louis Navier and George Gabriel Stokes in 1822, which are based on
mass and momentum conservation principles. The state of the fluid is completely
determined by the velocity field , tv r , pressure ,P tr and density , t r as a
function of position ,x x y y z zp p p p i i i where , ,x y zp p p are Cartesian
coordinates with unit vectors , ,x y zi i i and time t .
2.1.1 Single Phase Fluid Dynamics
In order to describe the dynamics of single fluid, the governing equations are NSEs,
based on the principles of mass and momentum conservation.
20
Mass Conservation Equation
The conservation of mass principle states that the mass can not be created or destroy.
So, the mass inside a fixed volume V changes only if mass flows in and out through
its boundary (Tryggvason et al., 2011). The differential form of continuity or mass
conservation equation is written as
0.t
v (2.1)
By using the definition of material derivative
,D
Dt t
v (2.2)
Moreover, expanding the divergence as, , v v v the convective
form of continuity equation is written as
,D
Dt
v (2.3)
For an incompressible fluid, the density is constant, so Eq.(2.3) becomes
0, v (2.4)
known as divergence-free or solenoidal condition of the velocity field v .
Momentum Conservation Equation
In the classical approach, the motion of a single fluid is studied through conservation
of linear momentum, which states that the rate of change of fluid momentum in fixed
volume V is the difference of momentum flux through the boundary plus the net
forces acting on the volume (Tryggvason et al., 2011). The differential form of
momentum equation is
,s bt
vvv f f (2.5)
or,
21
.s bt
vvv f f (2.6)
Where, vv is the dyadic product define as i jv vvv , sf , bf are surface and body
forces, respectively. The body force bf is the sum of all volumetric forces such as
gravitational force g f g and surface force sf is the sum of all forces acting on the
surface of the volume element of the fluid such as pressure pP f and viscous
friction τ f . The net pressure is derived as volumetric term P and viscous
friction as deviatoric stress tensor τ by using Gauss's divergence theorem.
Usually, both terms are combined into the stress tensor Tij ij ijP . By using the
definition of material derivative, continuity equation (2.1) and expanding the
divergence of nonlinear term as vv v v v v , Eq. (2.6) can be
written as
s b
D
Dt
vf f (2.7)
For compressible Newtonian fluid, the constitutive relation between the deviatoric
stress tensor and velocity field prescribed as (Tryggvason et al., 2011)
2
,3
T
τ v v v v (2.8)
where, and represent dynamic and bulk or volume viscosity, respectively. For
an incompressible fluid, the deviatoric stress is reduced as
.T
τ v v (2.9)
The final form of NSEs is obtained by inserting the surface and body forces into Eq.
(2.7)
2
.3
TP
t
vv v v v v v g (2.10)
For an incompressible fluid, Eq.(2.10) is reduced to
TP .
t
vv v v v g (2.11)
22
In order to solve aforementioned governing equations some additional constraints on
flow variables are needed on the boundaries of computational domain termed as
boundary conditions. There are different kind of boundary conditions, for example at
the inlet boundaries velocity fields are prescribed, and outlet boundaries are
considered far away from the region of strong flow variations so along the direction
of motion the change of variables can be neglected (zero gradient condition). Another
type of boundary is the solid wall boundary i.e., a contact surface of solid wall and
fluid. On that boundary, the no-slip and non-penetration conditions are applied for
velocity fields. By defining, the normal n̂ and tangential t̂ unit vectors on solid
wall (see Fig. 2.1) the no-slip condition is written as
ˆ ˆ,f w v t v t (2.12)
where, subscripts f and w stand for fluid and wall, respectively. This condition tells
that fluid does not slip on the wall because tangential fluid velocity ˆf v t on the
wall is equal to tangential velocity component ˆw v t of the wall. If the wall is
stationary then this condition leads to
ˆ 0.f v t (2.13)
Additionally, the normal component of fluid velocity ˆf v n is zero on the wall in
case the fluid is not flowing from or into the walls (non-penetration condition)
ˆ 0.f v n (2.14)
23
Fig. 2.1. Normal and tangential unit vectors on fluid/wall interface.
2.1.2 Two-Phase Fluid Dynamics
A variety of two-phase flows exists depending on combinations of two-phases as
well as on the interface structures. Two-phase mixtures are characterized by the
existence of the interface and discontinuities at the interface. The dynamics of two-
phase fluid flows are strongly influenced by the interfacial tension caused by the
interface that separates two phases. A schematic sketch of two-phase system is
shown in Fig. 2.2.
24
Fig. 2.2. Two-Phase fluid flow system separated by the interface 1,2 .
In order to analyze the two-phase fluid flow of two incompressible, immiscible fluids
the standard methods of continuum mechanics are valid. Thus, a two-phase flow is
considered, a field that is subdivided into single-phase regions with moving
boundaries between phases. The standard conservation equations such as
conservation of mass and conservation of momentum hold for each subregion with
appropriate boundary conditions to match the solutions of these equations at the
interfaces. The dynamics of two-phase fluid flows enforce that the continuity
equation, momentum equations and boundary conditions on fluid/solid wall surface
s must be solved independently for each phase in connection with boundary
conditions at the fluid/fluid interface 1,2 . To describe the motion of two fluids, the
governing equations in compact form are
0,i v (2.15)
Tii i i i i i i iP ; i = 1,2,
t
vv v v v g (2.16)
where t is the time, iv is the velocity, iP is pressure, and i stands for the dynamic
viscosity. i is the density and g is the gravitational force, respectively. The
subscript i is used to distinguish each phase. The no-slip boundary condition for
fluid/solid interface is prescribed as
,ˆ ˆ ,
s sf i w
v t v t (2.17)
25
and in case of stationary wall, Eq.(2.17) reduces as
,ˆ 0.
sf i
v t (2.18)
Additionally, non-penetration condition for stationary wall is prescribed as
,ˆ 0.
sf i v n (2.19)
Subsequently, the equations (2.15)-(2.16) can be applied to each phase or region up
to the interface, but not across it. A particular form of balance equation is used at
fluid/fluid interface in order to take into account the sharp changes or discontinuities.
Specifically, it is assumed that interface has surface tension, on which applying a
stress balance on the interface leads to the interfacial boundary condition
(Tryggvason et al., 2011)
1,21 2
ˆ ˆ , T T n n (2.20)
where, 1 2,T T represent stress tensor in each fluid, ˆ, n are surface tension and unit
vector normal points into the phase 1. is the mean curvature. In addition, an
interface between two immiscible fluids is impermeable and conservation of mass
across the interface leads to
1 2ˆ ˆ ,nv v n v n (2.21)
where, 1 2,v v represent velocities of both fluids and nv is the normal component of
the velocity of the interface. Finally, for the viscous fluids, the continuity of
tangential velocity across the interface (Tryggvason et al., 2011) leads to
1 2v v (2.22)
26
2.2 Phase Field Model
2.2.1 Sharp and Diffuse Interface
An important feature in two-phase system is the presence of interface and related
discontinuities at the interface. The nature of the interface between two fluids has
been the subject of extensive investigation in many scientific and engineering
applications for two centuries. The initial investigation by Young, Laplace and Gauss
(Young, 1805), considered the interface between two liquid as a zero thickness
(sharp interface) surface endowed with physical properties such as surface tension. In
these investigations, based on static or mechanical equilibrium, the physical
quantities such as density, viscosity are discontinuous across the interface (see Fig.
2.3). Physical processes such as capillarity occurring at the interface are represented
by the imposed boundary conditions. The equations of motion that hold in each fluid
are supplemented by the boundary conditions on the interface that involve the
physical properties of the interface as described previously in subsection 2.1.2 . The
major numerical problem is the implementation of the boundary conditions on the
moving interfaces and across the sharp interface where certain quantities may suffer
jump discontinuities. In order to overcome this difficulty, it was decided by Poisson,
Maxwell, and Gibbs (Gibbs, 1878), that the interface in fact represents a rapid but
smooth transition layer, where the physical quantities are smoothly changed between
the two different bulk values (see Fig. 2.3). This Diffuse Interface (DI) idea was
further developed in 1892 by Rayleigh (Rayleigh, 1892) and in 1893 by Van der
Waals; the later work was originally published in Dutch and then translated to
English (Rowlinson, 1979). A contemporary representative theory for the diffuse-
interface notion that had roots in Rayleigh and van der Waals ideas is the phase field
method (PFM) (Jacqmin, 1999).
2.2.2 Phase Field and Free Energies
In PFM, sharp interface is replaced by diffuse interface of finite thickness by
introducing phase field variable , to characterize the bulk phases and interfacial
region. The phase field variable has distinct constant values in each bulk phase such
as 1 in one phase and 1 in another phase and varies continuously over thin
27
interfacial region 1 1. Subsequently, it is easy to handle the numerical
computation of interface movement and deformation. For diffuse interface models,
the governing equations depend on how to model the total free energy of the system.
For an immiscible, Newtonian two-phase flow, the total free energy only consists of
the mixing energy of the interface.
Mixing energy: for an immiscible, Newtonian two-phase flow, molecular forces
determine the diffuse interface structure; the tendencies for mixing and demixing are
balanced by the non-local mixing energy. The general form of the mixing-energy
density (Jacqmin, 1999) as a function of and its gradient is as follows
2
0
1, ,
2mixF f (2.23)
where the first part corresponds to the interfacial energy, which represents weakly
non-local interactions between two phases that prefers complete mixing. The second
part represents the free energy density of uniform system that prefers complete
separation of the phases. is of order and is of order 1 and these
parameters leads to the interface width with thickness and surface tension
1 .The simplified form of Eq.(2.23), following (Cahn and Hilliard, 1958), is
written as
2
2
0, .2
mixF f
(2.24)
Where, is the magnitude of mixing-energy density and it is defined as x
(Liu and Shen, 2003). The second part 2
2
0
11
4f has two minima
corresponding to two stable phases, shown in Fig. 2.4. Finally, the total free energy
for two-phase flow is expressed as
, ,mixF F d
(2.25)
28
Fig. 2.3. (a) Discontinuous physical properties across a sharp-interface. (b)
Continuous physical properties across diffuse interface.
Fig. 2.4. Bulk free energy as a function of order parameter (see Eq. (2.24)).
29
2.2.3 Cahn-Hilliard Dynamics
Fick’s law states that the diffusive flux of particles in a system is directly
proportional to the gradient of concentration and that the particles can’t be created,
destroyed or switched. Based on Fick’s law, the Cahn-Hilliard (CH) (Cahn and
Hilliard, 1958) equation is obtained for phase field variable as
.t
j (2.26)
CH (Cahn and Hilliard, 1959) extended the Van der Waal’s idea to time-dependent
situation with assumption that diffusion occurs by minimizing the free energy and
interfacial diffusion fluxes are proportional to the gradient of chemical potential
(Jacqmin, 1999)
,M j (2.27)
where, ,M are mobility and chemical potential respectively. Based on free energy
(2.25) the chemical potential is obtained by variational derivative of free energy
(Euler-Lagrange equation) and is defined as
.mix mixF FF
(2.28)
By considering the mixing energy in Eq.(2.24), the chemical potential can also be
rewritten as
3 2 2
2.
(2.29)
Finally, the CH equation (2.26) leads to
2
3 2 2
2.
Mt
(2.30)
Which is time-dependent, highly non-linear 4th order PDE but is decoupled into two
2nd order PDEs.
30
2.2.4 Surface Tension and Interface Width
As mixing energy mixF is concerned with the molecular interaction of two phases so
it contains the classical concept of interfacial tension. (Yue et al., 2004) derived a
relationship between the parameters of Eq. (2.24) and an interfacial tension , which
not only indicates the connection to the sharp-interface limit but also gives a rule to
translate the parameters into sharp interface. By considering one-dimensional
interface, (Yue et al., 2004) required that diffuse mixing energy in a region be equal
to the surface energy
2
0
1.
2x
x
df dp
dp
(2.31)
By assuming that the diffuse interface is at equilibrium, the chemical potential has to
be equal to zero
2
020.
x
df
dp
(2.32)
By multiplying equation (2.32) with / xd dp and integration leads to
2
0
1,
2 x
df
dp
(2.33)
gives equal partition of free energy between two terms in an equilibrium. An
equilibrium profile for x can be obtained using boundary condition 0 0 by
integrating the equation (2.33) as
tanh / 2 .x xp p (2.34)
Which represents two stable uniform solutions 1xp for bulk phases and for
non-uniform interfacial region in the interval 1, 1 , as shown in Fig. 2.5. The
interface width measures the thickness of the diffuse interface. According to (Yue
et al., 2004), 90 % of variation in occurs over a thickness of 4.1641 , while 99 %
31
of the variation relates to a thickness of 7.4850 . The profile for given in
Eq.(2.34) gives the absolute minimum of the free energy, which contains many local
minima corresponding to a family of periodic profiles (Mauri et al., 1996). By
substituting Eq. (2.34) into Eq. (2.31), the value for surface tension can be
evaluated as
2 2
.3
(2.35)
As the interfacial thickness tends to zero, so should the energy density parameter
; their ratio gives the interfacial tension in the sharp-interface limit. A detail proof
of the diffuse interface model converging to the conventional Navier-Stokes system
with sharp interface can be found in (Liu and Shen, 2003).
Fig. 2.5. Hyperbolic tangent profile for plane interface at an equilibrium.
2.3 Hydrodynamic Coupling
In this section, the hydrodynamic coupling of NSEs and CH equation is presented.
An important issue while handling the time-dependent two-phase flow due to a priori
unknown position of the interface is how to tackle the interfacial tension. This issue
has been investigated by many researchers in (Anderson et al., 1998; Brackbill et al.,
1992; Scardovelli and Zaleski, 1999). In the classical approach, an interface between
two incompressible, immiscible fluids is considered as a sharp interface. In case of
32
sharp interface, the interfacial tension across the interface is represented by the
imposed boundary conditions on the interface. The compact form of governing
equations as well as interface boundary conditions in the classical sense are depicted
in Fig. 2.6.
Fig. 2.6. Incompressible Navier-Stokes equations for two-phase flow together with
interface boundary conditions for sharp interface.
On the other hand, in diffuse interface model or PFM, the interface evolution is
governed by CH equation. Subsequently, the NSEs are modified by adding the
phase-field dependent surface tension force stf , representing interfacial tension.
However, these modified NSEs are solved for the whole fluid domain as phase field
variable adjusts the material properties and indicates each bulk phase. According
to (Jacqmin, 1999), the basic ideas for deriving the diffuse interface fluid dynamical
force are mentioned as follows:
The amount of free energy can be changed through convection either by
lengthening, thickening/thinning interfaces.
There must be a diffuse interface force exerted by the fluid such that the
change in kinetic energy is always opposite to the change in free energy.
Rate of change of free energy due to convection leads to
33
.conv conv
F Fd
t t
(2.36)
As it has been given in (Villanueva, 2007) that,
,
.
convt
F
v
(2.37)
Then, integrating by parts and using the identity v v v
Eq.(2.36) yields
,
.
conv
Fd
t
d d
v
v v
(2.38)
As both fluids are incompressible, so by taking into account the divergence-free
constraint 0 v Eq.(2.38) becomes
.conv
Fd
t
v (2.39)
Furthermore, rate of change of kinetic energy due to surface tension is always
opposite to the change in free energy
.st
conv kinetic
F Ed
t t
v f (2.40)
For the two to be equal and opposite for arbitrary and v , it must be true that
.st f (2.41)
So the modified momentum equation for two immiscible, incompressible fluids can
be written as follows
,T
st bP t
vv v v v f f (2.42)
34
where, density and viscosity are the functions of phase field variable as
1 2
1 2
1 / 2 1 / 2,
1 / 2 1 / 2.
(2.43)
Here, it is needed to remark about surface tension force stf as different forms of
surface tension force exist in the literature. In (Anderson et al., 1998; Feng, 2006; Liu
and Shen, 2003), the phase-field variable dependent surface tension force term
is used. Recently in (Kim, 2005), different forms of surface tension
force are summarized as follows
26 2 ,st I f (2.44)
6 2
,st
f (2.45)
6 2
.st
f (2.46)
Furthermore, new form of surface tension force is proposed in (Kim, 2005) as
2
6 2st
f (2.47)
2.3.1 Phase-Fields Models for Large Density and Viscosity Ratios
In this subsection, the PFM for large density and viscosity ratios derived by (Ding et
al., 2007) is presented. Model H (Gurtin et al., 1996) has gained much popularity for
the simulation of two immiscible, incompressible density matched fluids. Model H
consists of continuity and momentum equations for a divergence free velocity field
together with convective CH equation. This model has been used by (Jacqmin, 2000)
and (Ding and Spelt, 2007) for the analysis of the flow near a moving contact line
and compared the results with sharp interface method. Afterwards, Model H has been
modified by replacing constant density with variable density together with
divergence-free velocity field. This so-called modified H model has become an
appealing computational method for two-phase flow due to the smooth variation of
the order parameter across the interface. This modified H model has been used for
the simulation of Rayleigh-Taylor instability (Jacqmin, 1999) and for flows with
35
moving contact lines (Ding and Spelt, 2007; Jacqmin, 2000, 2004). A quasi-
incompressible diffuse interface model has been derived by (Antanovskii, 1995) by
assuming that the immiscible liquids can mutually penetrate into each other in such a
way that the sum of the mass diffusive flow rates of the two fluids equals zero. As a
result he obtained the conventional compressible continuity equation
0,t
v (2.48)
such that the velocity is divergence free only for equal bulk densities. (Ding et al.,
2007) derived the convective CH equation and continuity equation for a divergence
free velocity with the assumption of incompressibility of the two-fluid mixture.
Consider the flow of two incompressible, immiscible fluids of different densities
1 2, and viscosities 1 2, . The volume fraction *C ( *; 0 1C C is equivalent to
1 / 2 with phase field variable ) of one fluid is used to represent the
composition of the two components in a volume element in the domain. The local
densities of both fluids are as follows
* *
1 1 2 2, 1C C (2.49)
Then the local average density is
* *
1 21C C (2.50)
The mass conservation for fluid 1 in the bulk region can be written as
11 0,
t
m (2.51)
where, 1m represents the mass flow rate (per unit volume). In the bulk region, only
advection is considered to the mass flow such as 1 1m v whereas in the interfacial
region between two fluids, a smooth transition of *C is preserved by diffusion and
contribution of diffusive flow is also considered in the total mass flux. The diffusive
mass flow for fluid 1 is represented by 1 1 j , where 1j is the volume diffusive flow
rate. Then the mass conservation for fluid 1 in the interfacial region can be written as
11 1 1 0.
t
v j (2.52)
36
Similarly, for fluid 2
22 2 2 0.
t
v j (2.53)
The substitution of the density expression from Eq.(2.49) into Eq.(2.52) gives
*
*
1 0,C
Ct
v j (2.54)
for fluid 1 and similarly for fluid 2 gives
*
*
2
11 0.
CC
t
v j (2.55)
From Eqs.(2.54) and (2.55), we have
1 1 2 2 ,D
Dt
v j j (2.56)
and
1 2 v j j (2.57)
According to (Antanovskii, 1995) mass diffusive flows satisfy 1 1 2 2 0 j j , which
leads to Eq.(2.48). Hence, v is defined as mass average velocity, i.e., such that
1 1 2 v m m which is logically connected to the velocity in NSEs. In quasi-
incompressible diffuse interface model, the volume diffusive flow rates differ for
different bulk densities, and the total volume occupied by each fluid does not
conserve and the CH equation is also not exactly recovered. In (Ding et al., 2007),
the diffusive flow rate is not related to the densities but the local compositions of the
two components which leads
2 1, j j (2.58)
in the spirit of CH model and velocity is defined as a volume averaged velocity such
that 1 1 2 2/ / v m m . The volume diffusive flux of the two fluids in Eq. (2.58)
are of equal magnitude but in opposite direction. It is therefore easy to introduce
37
1 2, j j j j . Substitution of Eq. (2.58) into Eq. (2.57) gives the continuity
equation for divergence free velocity as follows
0, v (2.59)
and the evolution equation for volume fraction *C is obtained by substituting Eq.
(2.59) into Eq. (2.54) or Eq. (2.55) as follows
** ,
CC
t
v j (2.60)
known as convective CH equation. The diffusive flow rate j is directly proportional
to the gradient of the chemical potential (see subsection 2.2.3 ).
2.3.2 Boussinesq Approximation Model
Another way to model the two-phase system is the classical Boussinesq
approximation. In this approach, the density is treated as constant and difference of
actual and constant density contributes as a buoyancy force in the momentum
equation. In (Liu and Shen, 2003), the coupled NSCH Boussinesq model has been
used to analyze the dynamics of two-phase flow
*
0,
.T
buoP t
v
vv v v v f
(2.61)
2
3 2 2
2.
Mt
v
(2.62)
Where, *
1 2 / 2 is constant density and buof is the buoyancy force and
defined as
*
1 2 2 11 1 * .buo
f g g (2.63)
Boussinesq approximation is only valid if the difference of densities of both fluids is
not large. In the present work, the governing equations given in Eqs. (2.42)-(2.43)
and (2.60) together with continuity equation (2.59) for divergence free velocity
38
would be used for incompressible, Newtonian two-phase flow using two different
form of surface tension forces (see Chap. 4. and Chap. 5. ). In (Lee and Kim, 2012),
different form of Boussinesq approximation model with constant and variable density
has been used for buoyancy driven-flows, which would also be used for Rayleigh-
Taylor instability problem (see Chap. 4. ).
Equation Chapter (Next) Section 1
39
3. Numerical Method
Meshless methods are numerical algorithms; that use a set of arbitrarily distributed
nodes in the domain as well as on the boundary for the solution of physical
phenomena. The purpose of this chapter is to describe the DAM (Nayroles et al.,
1991) to solve free and moving boundary problems. The characteristics of the
meshless method are presented. The concept of collocation and Weighted Least
Square (WLS) for calculation of unknown coefficients is presented. Furthermore,
explicit time discretization, upwind scheme and pressure velocity coupling are also
discussed.
3.1 Characteristics of Meshless Methods
The development of simple and efficient algorithms for solving complex PDEs that
govern the physical phenomena is of great interest in engineering and applied
sciences. The extensively used numerical methods are FDM, FVM, and FEM. In
these classical methods, the computational domain of the problem is discretized into
polygons. Each computational domain is discretized by nodes and the mesh consists
of node positions along with topological information on connectivity. On the other
hand, in MMs some form of predefined mesh is still needed but the amount of
additional information about connectivity and topological relations between the
nodes is significantly reduced. The nodes can be arbitrarily scattered within the
computational domain and on the boundary. Although, classical methods such as
FVM, FEM, FDM and BEM are implemented efficiently and robustly for many fluid
dynamic problems but there are some limitations for these methods, namely:
Mesh generation of computational domain is prerequisite.
40
FDM requires uniformly distributed nodes.
Difficult to model free and moving boundaries.
Computational cost of re-meshing, for 3D problems, at each time step is
very expensive.
Meshless methods, also called mesh reduction or meshfree methods, generate system
of algebraic equations for the whole computational domain and boundary without
polygonisation (Atluri and Shen, 2002a; Liu, 2002; Liu and Gu, 2005). The MMs are
characterized by the following features:
No polygonisation is needed.
Complicated geometry is easy to cope with.
Accurate and efficient.
3.1.1 Domain and Boundary Discretization
The shape of the problem domain is very complex in real world. In addition, the
complexity and intensity of the included physical process can also be high. In order
to cope with physical and geometrical complexities, the domain should be discretized
as accurately as possible by finite number of nodes. Generally, the geometry is
simplified to a reasonable representation because of the constraints on the
computational resources and time.
The most common types of space discretization arrangements for numerical methods
are shown in Fig. 3.1. FEM discretization, which includes polygonisation with
triangles is shown in Fig. 3.1(a). These triangles can be of arbitrary orientation and
dimensions and can be interchanged with other kinds of polygons. The FVM
discretization includes polygonisation with rectangles, which are restricted to
coordinate directions (see Fig. 3.1). The FDM discretization depends only on
pointisation and points should be uniform and restricted to coordinates directions (see
Fig. 3.1(d)). One of the main reason for the development of new numerical method is
the complexity of mesh generation.
In case of Dual Reciprocity Boundary Element Method (DRBEM) (Partridge and
Brebbia, 2012), the domain discretization depends on pointisation instead of
polygons (see Fig. 3.1(c)) whereas boundary is discretized with straight lines. This
41
method belongs to the class of so-called mesh reduction or semi-meshless methods as
the domain discretization is replaced with pointisation. An important point of the
DRBEM is the approximation of the field in the domain by a set of global
approximation functions and subsequent representation of domain integrals of these
global approximation functions by the boundary integrals. However, the solution of
these boundary integrals suffers from cumbersome evaluation of regular, strongly-
singular, weakly-singular and hyper-singular integrals. Moreover, boundary
polygonisation is still needed. Consequently, the development of numerical methods
is tending towards complete meshless methods.
The discretization of MMs is shown in Fig. 3.1(e). A number of nodes are used to
discretize the domain as well as the boundary of the problem. Computational nodes
can be arbitrarily spaced and non-uniform. In MMs, the numerical solution can be
obtained by the construction of shape functions without any predefined knowledge of
geometrical connection between nodes. In the recent years, the most extensively used
methods for constructing the meshless shape functions are Weighted Least Square
(WLS) approximation and interpolation techniques.
Fig. 3.1. Discretization of geometry for different numerical methods: (a) FEM, (b)
FVM, (c) FDM, (d) DRBEM and (e) MSM (reproduced with the permission of
Springer eBook publication).
42
3.2 Node Distribution and Local Subdomain
The computational domain consists of N nodes, of which N nodes are positioned
inside the domain and N on the boundary. The computational domain is divided in
such a way that each node has its own local subdomain, consisting of its neighboring
nodes. Each local subdomain consists of arbitrary spaced l N nodes. In order to solve
PDEs at each computational node, the local subdomains need to overlap, which
means that the central node in one local subdomain is the calculation node of the
other local subdomain. A scheme of the global domain and overlapped local
subdomains for interior and boundary computational node is depicted in Fig. 3.2.
Fig. 3.2. Scheme of the discretization with the illustration of subdomains for the boundary
1loc and the domain computational nodes 2loc .
3.3 The Approximation Function
The approximation function p is introduced as a linear combination of basis
functions i p in the local subdomain l over arbitrary located nodes
; 1,2,...,l k lk Np in the following way
1 1
,b bN N
l i l i l i i l
i i
p p p p (3.1)
43
where, , , andl i b l i x x y y z zN p p p p i i i are unknown coefficients, number of
basis functions, basis functions and position vector. Different types of basis functions
can be used in the formulation of approximation functions. In the field of MMs, the
commonly used are polynomials, multiquadrics, inverse multiquadrics and Gaussian
functions.
3.3.1 The Collocation
The unknown coefficients lα in Eq. (3.1) are determined from the collocation
condition by keeping the number of nodes in local subdomain equal to the number of
basis functions b lN N . The collocation is defined as
,l k l k p (3.2)
where, l k are the corresponding data values for all nodes in local subdomain. By
considering the collocation condition for all local subdomain nodes leads to a
xb bN N system of equations as
l C l l A α (3.3)
The method is known as LRBFCM (Vertnik and Šarler, 2006), when applied to
numerically solve PDEs with local collocation with Radial Basis Functions (RBFs).
3.3.2 The Weighted Least Square Approximation
In order to determine locally smooth and differential approximation of the discrete
data on overlapping subdomains, the WLS method can also be used. The form of the
approximation is the same as in Eq.(3.1). For this purpose, the least squares problem
is formulated in terms of finding the minimum of the squared residuals l I as
2
1 1
.l bN N
l k l i i k l
k i
I
p p (3.4)
The WLS form of Eq. (3.4) is written as follows
44
2
1 1
.l bN N
l k l k l i i k l
k i
I
p p p p (3.5)
Where, k l p p is the weight function and will be elaborated in later subsection.
The minimization of Eq.(3.5) leads to a xb bN N system of equations to calculate the
unknown coefficients at each node kp of local subdomain l as
.l W l lA α (3.6)
In the present work, the WLS approximation with two-dimensional polynomial and
Gaussian weight function is used.
3.4 Spatial Discretization of Partial Differential
Equations using Diffuse Approximate Method
This section deals with the spatial discretization of PDEs using DAM.
3.4.1 Construction of Local Interpolant using Polynomials
Generally, by considering the computational domain with boundary the
following PDE needs to be solve
,
, ,t
tt
pp (3.7)
Where, is differential operator. The PDE given in Eq. (3.7) is solved subject to the
following initial conditions,
0, ; ,t p p (3.8)
and Dirichlet boundary conditions on D , Neumann boundary conditions on N and
mixed (Robin) boundary conditions on R , where D N R
45
, ; ,
,; ,
ˆ
,; .
ˆ ref
D D
N N
R R R
t
t
t
p p
pp
n
pp
n
(3.9)
Where, , , andref
D N R R
are known functions. The approximant p at
node p is defined as
1
,bN
l i i l
i
p p p (3.10)
where, l is the index of the subdomain ,l respectively. A two-dimensional
monomials are used as basis functions 6bN defined as
1 2
3 4
5 6
2
2
1, ,
, ,
, .
l l x xl
l y yl l x xl
l x xl y yl l y yl
p p
p p p p
p p p p p p
p p p p
p p p p
p p p p
(3.11)
The unknown coefficients l i are calculated by minimization of the expression
2
1 1
2
1 1
2
1 1
1
1
1
ˆ
l b
l b
l b
b
refkl
b
N N
l k l k l i i k l
k i
N ND
k l k l i i k l
k i
N NN
k l k l i i k l
k
k
D
k
N
k
i
NR R
k l i i k lNi
k l
R
k Nk
l i
i
I
p p p p
p p p p
p p p pn
p p
p p
2
ˆi k l
p p
n
(3.12)
where, , , andD N R
k k k k are domain and boundary indicators defined in
(3.13).
46
1; 1;
0; 0;
1; 1;
0; 0;
Dk D k
k k Dk k
N R
N Rk k
k kN R
k k
p p
p p
p p
p p
(3.13)
The minimization of Eq. (3.12) leads to the following xb bN N system of equations
1
; 1,2,...,bN
l ji i l j b
i l
A b j N
(3.14)
Where, the explicit form of the matrix l jiA and adjacent vector l jb for each thl
calculated node is given as
1
1
1
1
ˆ ˆ
ˆ
xˆ
l
l
l
l
N
l ji k l i k l j k l
k
N
k l i k l j k l
k
N
k l i k l j k l
k
N
k l i k l i k l
k l
D
k l
N
k l
R R
k
j k l
k l k
R
k j k l
A
p p p p p p
p p p p p p
p p p p p pn n
p p p p p pn
p p p pn
(3.15)
1
1
1
2
1
ˆ
ˆ
l
l
l
refkl
refk
N
l j k l j k l k
k
ND
k l j k l k
k
NN
k l j k l k
k
R R
k j k lN
k
k l
D
lR R
kk
k l
N
k l
R
k l
j k l
b
p p p p
p p p p
p p p pn
p p
p pp p
n
(3.16)
The unknown coefficients can be achieved by inverting the non-singular matrix l jiA
with defined elements in Eq.(3.15). The necessary condition for non-singular matrix
is that the number of nodes in the local subdomain l is equal or larger than the
number of basis functions l bN N .
47
3.4.2 Calculation of Differential Operators
The basic idea of DAM is to use the polynomial functions to estimate any linear
differential operator acting on a physical field p . As the coefficients l i are
constant, so
1
bN
l i i l
i
p p p (3.17)
The coefficients l i can be evaluated by solving the system given in Eq.(3.14) by
calculating the inverse 1
l A. Then, the expansion coefficients can be expressed in
terms of the components of l b using the inverse. So, Eq.(3.17) yields
1
1 1
,b bN N
l k l ik i l
k i
b A
p p p (3.18)
defining the multiplication of the field values with the operator coefficients.
Furthermore, Eq.(3.18) can be written as
1
bN
l k l k
k
b
p (3.19)
Where, l k represents operator coefficients defined as
1
1
.bN
l k l ik i l
i
A
p p (3.20)
3.4.3 Weight Function
One important choice to be made when using DAM is the weight function , which
determines the relative impact of nodes in Eq.(3.12). It has the peak value of 1 at the
central node of the subdomain l and decays with increasing Euclidean distance
from the central node lp . In each subdomain l , the influence of neighboring nodes
kp on the central node lp is expressed in terms of weight function, which has strong
48
influence on the stability, accuracy and condition number of matrix l A in Eq.(3.14).
Generally, weight function has to satisfy the following conditions
1. 0;l k l l p p p over l
2. 0;l k l l p p p outside l
3. l is a monotonically decreasing function
First condition is the positivity, which is important to ensure the stable representation
of the physical phenomena (Liu, 2002) but not a mathematical requirement. The
second condition is related to the compactness which enables the approximation to be
generated from the neighboring nodes. The last condition is imposed on the physical
consideration that the nodes in the vicinity have more influence than the more distant
nodes, but again is not a mathematical requirement.
The choice of the weight function for local numerical methods is more or less
arbitrary as long as the weight function satisfies the conditions of positivity and
compactness. For DAM, explicitly, it was proven that Gaussian function performs
better over the other weighting functions (Sophy et al., 2002). Respectively, Gaussian
weight function has been employed in the present dissertation as follow
2
2exp ;,
0 ;
kr
k k lh
k l
k k l
c r h
r h
p pp p
p p
(3.21)
where, k lp p is the Euclidian distance between nodes kp and lp , c is user
defined free parameter, which determines how fast Gaussian function decays to zero.
h is the minimum distance between the central node lp and all other nodes in
subdomain l . In literature, different algorithms exist for finding the optimal value
of shape parameter c for radial basis functions (Mavrič and Šarler, 2015; Rippa,
1999). In DAM, however, the choice of the shape parameter is less sensitive than in
RBF collocation methods. The heuristically defined free parameter c , explicitly for
DAM, ranges from ln 100 4.6 (Prax et al., 1998) to 6.25 found in (Belytschko et
al., 1996) or almost 7 (Sadat and Couturier, 2000). The effect of shape parameter c
will be presented in later chapters.
49
3.4.4 Upwind Scheme
In fluid mechanics, the problem becomes non-self-adjoint due to the existence of the
convective term, for which a special treatment is needed to stabilize the numerical
solution. In classical numerical methods such as FDM, FVM and FEM, upwind
schemes are extensively used to stabilize the convective term. The same concept is
also needed in the MMs to achieve better stability in convection dominant flows. In
the present dissertation, to stabilize the convective terms in momentum and CH
equation (see Chap. 2. ). An upwind approach (Lin and Atluri, 2001) is used, which
is introduced by the shift of Gaussian function and the evaluation point in the
opposite direction of the velocity as
.ll l
l
v
p pv
(3.22)
Where lp is the shifted central node of the subdomain and is the absolute size of
the shift that depends on the dimensionless Péclet number Pe and the size of the
local subdomain. The subdomain of the shifted position has the same computational
nodes as the original subdomain. The upwinded advection operators differ in two
aspects from the non-upwinded operators. The first difference is the change of the
weight function due to the shift of the Gaussian center, shown in Fig. 3.3. The second
difference is that the position, where the upwinded operators are evaluated, is also
shifted by replacing lp by lp in Eqs.(3.12)-(3.15).The adaptive shift size was first
proposed in (Lin and Atluri, 2001) for meshless local Petrov-Galerkin (MLPG)
method as
1 1
coth Pe ,2 Pe
h
(3.23)
where the Péclet number is defined as
Pe ,2
h
D
v (3.24)
where v is the magnitude of velocity, and D is the diffusivity coefficient.
50
Fig. 3.3.Scheme of central and upwind Gaussian weight function. The dots
represent the local subdomain. The blue curve is the original Gaussian center and
the green curve represents the upstream shifted Gaussian center.
3.5 Time Discretization
Moving boundary problems are transient problems. Such problems are space- and
time- dependent so discretization in space and time is required. The main purpose of
the discretization is to replace the derivatives with difference expressions and to
obtain the algebraic equations. The most common and widely used time
discretization methods are Euler methods such as explicit, implicit and semi-implicit.
The explicit time discretization is used in this dissertation.
3.5.1 Explicit Euler Time Discretization
In an explicit scheme, the values of dependent variables at 0t t are calculated
from the known variables at 0t . As each difference equation involves one unknown
the resulting algebraic equations at 0t t can be evaluated independently to obtain
the values of unknowns. Such scheme is easy to implement and parallelize and has
low computational cost for each time step. The major drawback of this scheme is its
conditional stability. Special care is required while choosing the time step, which can
51
become impractically small, as in the case of stiff problems. In order to make sure
that the explicit scheme is stable, both Courant-Friedrichs-Lewy (CFL)
1,x
t
p
v (3.25)
and the von-Neumann stability
2
1,
2x
D t
p
(3.26)
must be satisfied, where D is the diffusivity coefficient. The explicit time
discretization scheme is schematically depicted in Fig. 3.4.
Fig. 3.4. Illustration of explicit time discretization scheme.
3.6 Pressure-Velocity Coupling
In the solution procedure of incompressible fluids, an important part is the special
treatment of the momentum equation as the pressure is not included explicitly in the
continuity equation so special treatment is need for pressure-velocity coupling. The
extensively used numerical algorithm for pressure-velocity coupling is Semi-Implicit
Method for Pressure Linked Equations (SIMPLE) (Ferziger and Peric, 2012) and its
various modifications like SIMPLER (Latimer and Pollard, 1985), Pressure Implicit
with Splitting of Operators (PISO) (Jang et al., 1986). Furthermore, the local
Pressure-Velocity (PV) coupling algorithm based on SIMPLE has been used for
52
Darcy flow (Kosec and Šarler, 2008) and phase change problems (Kosec and Šarler,
2009).
Apart from aforementioned algorithms, an alternative option are so called projection
methods introduced in the late 1960s by Chorin (Chorin, 1968) and Temam (Temam,
1968), where Helmholtz-Hodge decomposition (Petronetto et al., 2010) is used to
decouple the pressure and velocity. The projection methods can be categorized into
three groups: the velocity correction, pressure correction and the consistent splitting
methods (Guermond and Shen, 2003a). In projection methods, two intermediate time
sub steps are carried out in each time iteration. In the first time sub-step of velocity
correction scheme (Guermond and Shen, 2003b), the viscous term is ignored. In the
second time sub-step, velocity is corrected by the independently calculated viscous
term. The pressure-correction schemes can be further divided into non-incremental,
incremental and rotational forms. The first work developed by Chorin (Chorin, 1968)
was non-incremental where velocity is calculated without the pressure term in the
first time sub step. In the second step, the velocity is corrected using the calculated
pressure gradient also known as Fractional Step Method (FSM).
It is non-trivial task to achieve high order time accuracy in the numerical
approximation of NSEs using fractional step projection methods. The basic feature of
this method is the decoupling of advection and diffusion from the incompressibility
condition, with the introduction of time splitting error in the computed solution. In
general, the splitting error depends on the size of the time step t and is independent
of how accurately the subproblem of each partial step is approximated. For instance,
the time splitting error is of t in non-incremental projection method (Chorin,
1968; Temam, 1968). Later, in order to get accuracy of 2
t or higher, an
incremental pressure-correction method was proposed by Goda (Goda, 1979) using
finite difference method to take into account the previous value of pressure.
Furthermore, it has also been analyzed for viscous incompressible flow using the
combination of Marker-And-Cell (MAC) and Crank-Nicolson (Van Kan, 1986).
An incremental pressure-correction scheme (Goda, 1979; Van Kan, 1986) is used for
pressure-velocity coupling in this dissertation. In this scheme, an intermediate
velocity *v is calculated from the momentum equation as
53
0
0* 1 1,
t
Tt st bt P
f fv v v v v v (3.27)
where, 0tv is the velocity at the beginning of time step and t is the time step size,
stf is the surface tension force dependent on the phase field variable for two-phase
flow and bf is the body force. The pressure correction is evaluated from the
Poisson equation by using the divergence-free constraint as
2 *, v (3.28)
subject to the Neumann boundary condition for as
0*ˆ ,ˆ
t
n v v
n (3.29)
where n̂ is the outward normal. Finally, the pressure and velocity are updated as
0 * ,t t
v v (3.30)
00 .tt t
P Pt
(3.31)
3.7 Description of the Solution Procedure
In this section, the complete solution procedure to analyze the dynamics of two-phase
flow is elaborated. We seek the solution of the velocity field v and phase field
variable at time 0t t by assuming the known values of v , and P at time 0t
and known initial and boundary conditions. The following explicit numerical method
is employed at each time step:
Step-I
The phase-field variable is calculated first, using CH equation as
54
00 0
0
2
3 2 2
2
,
.
tt t t
t
t M
v
(3.32)
Step-II
Once the phase field variable is calculated then phase-field variable dependent
density and dynamic viscosity is updated as
0
0 0
1 2
1 2
1 / 2 1 / 2 ,
1 / 2 1 / 2 .
t t ot
t t t
(3.33)
Step-III
Using updated density and dynamic viscosity, an intermediate velocity *v is
calculated from the momentum equation as
0
0* 1 1,
t
Tt st bt P
f fv v v v v v (3.34)
Step-IV
Subsequently, the pressure correction is evaluated from the Poisson equation by
using the divergence-free constraint as
2 *, v (3.35)
subject to the Neumann boundary condition for as
0*ˆ .ˆ
t
n v v
n (3.36)
where n̂ is the outward normal. In case of pressure correction, the locality of the
discretization is reflected in a sparse matrix, solved by using PARDISO solver
(Schenk and Gartner, 2004), which is available in Intel Math kernel (MKL) library
and is specialized for solving large sparse linear system of equations on shared
memory multiprocessors.
55
Step-V
Finally, the pressure and velocity are updated as
0 * ,t t
v v (3.37)
00 ,tt t
P Pt
(3.38)
and solution is ready for the next time step. The time stepping is stopped for
0 maxt t t , where maxt is a predetermined time. The block diagram of elaborated
algorithm is schematically shown in Fig. 3.5.
Fig. 3.5. Block diagram of the solution procedure.
56
3.8 Numerical Implementation
The numerical implementation of the aforementioned solution procedure is coded in
Fortran 2008 (Fortran, 2014) programming language in double precision. The
LAPACK library is used to solve system of equations by QR decomposition. Parallel
Direct Sparse Solver (PARDISO) (Schenk and Gartner, 2004) from the Intel Math
Kernel Library (MKL) is used to solve the sparse matrix. The numerical code is
parallelized using Open Multiprocessing (OpenMP) (OpenMP, 2013) library.
The main code is based on the EDO_SP library developed in (Košnik et al., 2017)
and upgraded in the present work. Matrix Laboratory (MATLAB) (Gilat, 2009) and
Paraview (Henderson, 2005) are used to plot the line and contour plots, respectively.
Equation Chapter (Next) Section 1
57
4. Rayleigh-Taylor Instability
Problem
The aim of this chapter is to verify the numerical code for incompressible,
immiscible Newtonian two-phase flow. For this purpose, the benchmark Rayleigh-
Taylor instability problem is considered. Three physically different models are
considered for the simulation of this benchmark problem.The meshless results for 2D
RT instability problem, presented in this chapter, have already been published in
(Talat, Mavrič, Hatić, et al., 2018).
4.1 Rayleigh-Taylor Instability Problem
4.1.1 Problem Description and Literature Review
The fluid interface becomes unstable when a heavier fluid is placed over a lighter
fluid in a gravitational field. A perturbation of this interface has a tendency to
increase with time, producing a phenomenon known as Rayleigh-Taylor (RT)
instability. This phenomenon describes the entrance of the fluid with a higher density
into the fluid with lower density in the form of mushroom-shaped protrusions. The
RT instability phenomena was initially discovered by Rayleigh (Lord, 1883) and
after that applied to explanation of all accelerated fluids by Taylor (Taylor, 1950).
This instability has also been used to describe a wide range of problems, such as
inertial confinement fusion, supernova explosions and remnants, nuclear weapon
explosions oceanography and atmospheric physics (Lee et al., 2011) and was for the
first time numerically implemented by (Harlow and Welch, 1965).
The dynamic variables required to describe the motion of fluids are the velocity and
the pressure, which are highly sensitive to the density and the viscosity. Boussinesq
58
approximation (Boussinesq, 1903) is typically applied for buoyancy-driven flows
with small density variations. It has successfully been employed to obtain the
numerical solution of RT instability using Lagrangian-Eulerian vortex method
(Tryggvason, 1988) and a new model proposed for the development of the RT
instability in the Boussinesq limit, using concentrations of vorticity along the
interface (Aref and Tryggvason, 1989). Numerical simulation of RT instability of
inviscid and viscous fluids has been analysed in (Forbes, 2009). Many numerical
methods including boundary integral methods (Baker et al., 1984), front tracking
methods (Popinet and Zaleski, 1999), VOF method (Gerlach et al., 2006) and LS
method (Chang et al., 1996) have been used to analyse the RT instability.
Additionally, the dynamics of RT instability of two immiscible fluids in the limit of
small Atwood numbers together with surface tension effect has been numerically
analysed using PFM (Celani et al., 2009). In (Ding et al., 2007) a phase field
formulation has been developed with zero surface tension to analyse the RT
instability for large density variations. A long time simulation of the evolution of RT
instability (Lee et al., 2011) and a comparison of Boussinesq approximation and
variable density models on buoyancy-driven flows (Lee and Kim, 2012) has been
analysed using PFM.
4.2 Governing Equations
4.2.1 Problem Formulation
The geometry under consideration consists of a rectangular domain having width
L=1 m and height H=4 m. The RT instability problem in the present context, is a
system consisting of two immiscible incompressible fluids having different constant
densities and the same constant viscosities. The more dense fluid is placed above the
less dense fluid. A scheme of the problem is depicted in Fig. 4.1.
59
Fig. 4.1. Scheme of the geometry, initial conditions and the boundary conditions of
the Rayleigh-Taylor instability problem.
4.2.2 Model Formulation
The governing equations for unsteady, viscous, and immiscible two-fluid system are
the coupled NSCH equations. The general form of NSCH equations for two-phase
system is as follows
0,t
v (4.1)
2 2
3st bP
t
vvv v v f f (4.2)
2
3 2 2 ,
Mt
v ,
(4.3)
where t is the time, v is the velocity, P is pressure, and stands for the dynamic
viscosity. is the density and stf represents the surface tension force, determined by
60
the phase field model. bf stands for the body force, such as gravitational force with
2 2, ; 0m s , 9.81m s .x y x yg g g g g is the phase field variable with
1 representing different values in bulk phases, is the magnitude of the free
energy; 3 / 2 2 where is the surface tension. , ,M are the mobility, the
chemical potential and the interface width, respectively. Three physically different
phase field models are considered in order to simulate the RT instability problem. In
all three models, the CH equation is the same and both heavier and lighter fluids are
assumed incompressible and immiscible.
Model-I
In the present model, the formulation is the same as in (Dong and Shen, 2012) with
constant viscosity and variable density, including the surface tension effect appearing
in the momentum equation. The convective CH equation and the condition that the
velocity is divergence-free has been derived in subsection 2.3.1 . So, the Eqs.(4.1)
and (4.2) yield
0, v (4.4)
2 2 1,
2P
t
vv v v g (4.5)
where, 1 21 / 2 1 / 2 and last two terms in Eq.(4.5) represent
the surface tension and the body force, respectively.
Model-II
The present model is constructed by using the phase-field dependent density as in
Model-I and neglects the surface tension effect. This model is developed by adding
and subtracting the term * *
1 2, / 2 g from the momentum equation as in
(Lee and Kim, 2012). Eq.(4.4) is valid for this model and Eq.(4.2) yields
2 *
* .Pt
vv v v g g (4.6)
Eq.(4.6) can be further rewritten as
61
21 At1 ,
1 AtP
t
vv v v g (4.7)
with the rewritten buoyancy expression
*** 1 2
1 2
1 11 At2 21, ,
1 11 At 1 At
2 2
(4.8)
with Atwood number: 1 2 1 2At / .
Model-III
In this model, the Boussinesq approximation is used, where the density can be treated
constant in all terms except for the source term. The difference between the actual
density and the constant density * describes the buoyancy force in the
source term as in (Lee and Kim, 2012) and respectively, Eq. (4.4) is valid and Eq.
(4.2) yields
* 2 * * .Pt
vv v v g g (4.9)
Moreover, Eq.(4.9) is written as
2
* *
1(1 At ) ,P
t
vv v v g (4.10)
with the rewritten buoyancy expression
1 *
*
* *
1 1
2 2 At .
(4.11)
The differences between the models can be summarized as follows. Model-I includes
the surface tension effect while in Model-II and Model-III the surface tension effect
is neglected. In Model-I, the phase field dependent density is used to correctly
describe large density variations whereas small density variations are considered in
Model-II and Model III. Boussinesq approximation is used only in Model-III. Model
II represents variable density model for small density variations. The main reason for
considering three different models is to demonstrate the applicability of meshless
DAM for solving several different phase field models of RT instability.
62
4.2.3 Initial and Boundary Conditions
The governing equations are subjected to the initial and boundary conditions that
follow. Initially ot t , both fluids are at rest i.e.
0, , 0 m / sx y t v (4.12)
An initial profile, perturbed by sinusoidal wave of amplitude 0A , is considered for
phase field variable
0 0, , tanh 2 cos 2 / 2 .x y t y A x (4.13)
The no-slip boundary conditions are prescribed for velocity at the top and at the
bottom of the domain and the symmetry boundary conditions for the east and the
west side of the domain. The Neumann boundary conditions are defined for phase
field variable on all sides of the domain.
0, 0,
0, 0,
0, 0,
0, 0,
top top
bottom bottom
east east
west west
x y
x y
y
x
y
x
v
v
v
x
v
x
v
v
v
v
(4.14)
0, 0,
0, 0.
top bottom
east west
y y
x x
(4.15)
4.3 Results and Discussions
4.3.1 Sensitivity Study with Respect to DAM Parameters
In this section, a sensitivity analysis of the parameters of the numerical method is
performed. The effect of the shape parameter c and the number of computational
nodes in a local subdomain is analysed. For this purpose Model-I, given by Eqs.(4.3)-
(4.5) and subject to the boundary conditions Eqs.(4.14)-(4.15) is used. The initial
condition is given by a perturbed profile for phase field variable with amplitude
63
0.05 moA (4.13). The other parameters for the simulations with Model-I are given
in Table 4.1.
Table 4.1. Material Properties used in simulations with Model-I.
Material property Symbol Value
Viscosity 0.00313kg/(m s)
Interface width 0.01m
Mobility M 4 49x10 m / Ns
Surface tension 1.0 N/m
Magnitude of free energy 0.011N
Density of heavier fluid 1 31.225kg/m
Density of lighter fluid 2 30.1694kg/m
Three different values of the shape parameter 2.5, 5,10c , are used for 9 nodes in
local subdomain. The simulations are performed for two different node
arrangements 64x256 and 128x512 until 0.9st with time step 410 s and 510 s ,
respectively. Both Courant-Friedrichs-Lewy (CFL) (4.16) and von Neumann
stability conditions (4.17) are satisfied in numerical examples:
1,x
t
p
v (4.16)
2
0.25.x
D t
p
(4.17)
For all simulations, the time step is restricted by using
2
20.25 / ; / .xt p D D The results are shown in Fig. 4.2a and Fig. 4.2b. It
is observed that for the values of shape parameter 2.5 and 5.0 the results are almost
the same. The results for 10,c show that the heavy front moves faster than the
light one and both left and right tails have a significant bending curve. The
64
comparison of the present method with FVM is presented in more detail in
subsection 4.3.3 .
To study the node density convergence of the method, the solution at time 0.9 st
is compared for three different node arrangements 64x256, 128x512 and 192x786
with 10,c and with different number of nodes in local subdomain; i.e., 11 and
13. The results for three different node arrangements are presented in Fig. 4.3. It is
found that when using 11 and 13 nodes in local subdomain, the resulting shapes of
the interface are almost the same. There is a slight difference between the results of
node arrangement 64x256 and 128x512 but the results for node arrangements
128x512 and 192x786 overlap, which means that the solution for 128x512 is node-
density-converged.
65
Fig. 4.2. Model-I. Contours of RT instability for (left) 64x256 node arrangement and
(right) 128x512 node arrangement at 2.5, 5.0 and 10c for nine nodes in local
subdomain at 0.9 st .
66
Fig. 4.3. Model-I. Contours of RT instability for eleven (left) and thirteen (right)
nodes in local subdomain with 10,c for different node arrangements at 0.9 st .
4.3.2 Effect of Atwood Number on the Height of Bubbles and Spikes
In the RT instability, the fingers of the lighter fluid penetrate the heavier fluid as
bubbles, while the spikes of the heavier fluid move into the lighter fluid. To analyse
the effect of the Atwood number on the height of the bubbles and spikes, Model-II
and Model-III are considered with an initially perturbed profile Eq. (4.13) with
amplitude 0 0.1m.A The simulations are performed using 128x512 computational
node arrangement with time step 510 s using eleven nodes in local subdomain with
shape parameter 10c . The input parameters for the simulations of Model-II and
Model-III are given in Table 4.2.
67
Table 4.2. Material properties used in simulations with Model-II and Model-III.
Material property Symbol Value
Viscosity 0.01kg/ms
Interface width 0.01 m
Mobility M 40.1 m / Ns
Magnitude of free energy 1.0 N
The initial profile and the height of the bubbles hb and the spikes hs is measured as a
distance between the two tangent lines of tips of bubbles and spikes, respectively,
shown in Fig. 4.4.
Fig. 4.4. Left: Initial phase field variable distribution in the cavity with 0.1moA
Right: The definition of height of the bubbles and the spikes.
Time evolution of the interface at At 0.1, 0.3, 0.5 for Model-II using 11 nodes in
local subdomain with 10c is shown in Fig. 4.5. Similarly, the time evolution of
68
the interface for Model-III at At 0.5 using 11 nodes in local subdomain with
10c is shown in Fig. 4.6. It is observed that the symmetry of the heavier and the
lighter fluids is preserved for Model-III, and is not affected by the Atwood number.
The time of the snapshots in Fig. 4.5. are chosen such that the height of the bubbles
of the lighter fluid is the same for all the cases. The results for At 0.1 look
similar for both Model-II and Model-III. For At 0.3 and At 0.5 , the heavy
front moves faster than the light front.
Fig. 4.5. Time evolution of the interface for Model-II (a) At 0.1 , (b) At 0.3 ,
and (c) At 0.5 using 11 nodes in local subdomain with 10c .
69
Fig. 4.6. Time evolution of the interfaces for Model-III, At 0.5 using 11 nodes in
local subdomain with 10c .
The symmetry of the results is lost for Model-II although the flow starts with the
same symmetric profile. To illustrate this, the interfaces for both models are
compared in Fig. 4.7. Each plot contains two profiles. The shape of the original
interface is denoted with a solid line. The interface denoted by a dashed line is
obtained from the original interface by the following transformation. First, the
profile is mirrored over the x=0 axis and then again mirrored over the y=0 line. The
obtained profile is then shifted for L/2 to the right and plotted. The difference
between the profiles shown in Fig. 4.7a displays the effect of more accurate Model-
II. The simplifications involved in the derivation of the Model-III cause the
original and the shifted profile to coincide. The small differences between the
profiles in Fig. 4.7b are introduced by the numerical scheme.
70
Fig. 4.7. Left: Model-II. Right: Model-III. Interfaces for At 0.5 at 1.1s.t Both
simulations are done with 11 nodes in a subdomain and 10c .
The results for the height of the bubbles hb versus the height of the spikes hs at
At 0.01, 0.1, 0.3 and At 0.5 are shown in Fig. 4.8a with markers. The lines are
used to display the results given by (Lee and Kim, 2012). It is seen clearly that for
Model-II, the height of spikes is almost the same as the height of the bubbles for
small values of At but as At increases the height of the spikes decreases as
compared to the height of the bubbles. On the other hand, in case of Model-III, the
fronts of the light and the heavy fluid propagate with the same speed regardless of
the value of Atwood number. The results are found to be in excellent agreement
with the reference results (Lee and Kim, 2012).
71
Fig. 4.8b shows the time evolution of the interface for both models at At 0.5 at
1.1st . In both cases, the approximate value of hs is 1.2. On the other hand, the
approximate values of hb for Model-II and Model-III are 1.45 and 1.2,
respectively.
Fig. 4.8. Left: The height of the bubbles hb versus the height of the spikes hs for
At 0.01, 0.1, 0.3 and At 0.5 of Model-II and Model-III. The solid lines
represent the results (Lee and Kim, 2012) and markers show the present results.
Right: A comparison of inter-fluid boundary of Model-II and Model-III for
At 0.5 at 1.1s.t
4.3.3 Comparison with Finite Volume Method
Here the meshless results are compared with the results (Popinet and Zaleski, 1999)
obtained by open source code Gerris, developed by Popinet (Popinet, 2003, 2009).
72
Gerris solves the time-dependent incompressible Navier-Stokes equations using
second-order time and space discretization on Cartesian grids based on FVM. For
two-phase flow simulations, it uses VOF (Hirt and Nichols, 1981) and Piecewise
Linear Interface Construction (PLIC) (Youngs, 1982b) algorithm for interface
reconstruction. Local curvature on the interface is calculated by generalized height
function (HF) (Cummins et al., 2005). It controls the time step size and classical
Courant-Friedrichs-Lewy (CFL) condition to ensure the meaningful use of VOF
advection algorithm. One of the powerful features of Gerris is its capability of using
adaptive mesh refinement based on the octree (quadtree in 2D) division of cells. The
resolution is adapted to the features of the flow automatically and dynamically,
which enables extreme grid refinement in the vicinity of the interface. Furthermore, it
runs in parallel employing the MPI library, and the computational domain is
partitioned by utilizing the exiting box boundaries.
Because of this Model-I is considered with the same initial and boundary conditions
and the same input parameters as described previously in Table 4.1. The meshless
simulations are performed having 11 nodes in local subdomain and shape parameter
10c using 128x512 node arrangement. The comparison of the present results for
shape parameter 10c and 11 local subdomain nodes with the FVM results on 128
x 512 mesh is shown in Fig. 4.9. It is seen clearly that for these parameters of DAM
our simulations are in close agreement with the reference FVM results (Popinet and
Zaleski, 1999). Fig. 4.10 and
show the time evolution of the interface at different times with node arrangements
64 x 256 and 128 x 512, respectively. It is found that the surface tension changes the
shape of the mushrooms as compared to the results of Model-II and Model-III.
73
Fig. 4.9. A comparison of DAM (Model-I) and FVM results at different times
using 128 x 512 node arrangement. The solid and dashed lines represent FVM and
DAM results with 11 points in local subdomain and 10c , respectively.
74
Fig. 4.10. Model-I. Time evolution of the moving boundary by using 64 x 256
node arrangement with 11 nodes in local subdomain for 10c .
Fig. 4.11. Model-I. Time evolution of the interface by using 128 x 512 node
arrangement with 11 nodes in local subdomain for 10c .
Equation Chapter (Next) Section 1
75
5. Meshless Phase Field Method for
Two-Phase Flow
This chapter deals with the phase field simulations of two-phase flow together with
DAM. The numerical results, including study of mesh independence, comparison
with FVM results and effect of process parameters on results are presented. The
results, presented in this chapter, are currently under review (Talat, Mavrič, Belšak,
et al., 2018).
5.1 Governing Equations
5.1.1 Problem Formulation
The two-phase problem under consideration is as follows. We deal with a co-flow
microfluidics problem, consisting of two coaxially aligned capillary tubes. The tubes
have inner tube radius 10μm,iR inner tube length 20 μm,H outer tube radius
30μmoR and outer tube length 400 μm.L An incompressible, Newtonian fluid
with density i and viscosity i flowing at a constant flow rate ,iQ is injected
through a capillary tube of radius iR into a co-flowing immiscible, incompressible
Newtonian fluid having density o and viscosity ,o flowing at a constant flow rate
oQ . The outer fluid in contained in a coaxial cylindrical tube of radius oR . The
thickness of the inner tube wall is negligible. The geometry of the problem of interest
is depicted in Fig. 5.1. This arrangement is a prototype for flow focusing.
76
Fig. 5.1. Diagram scheme of the geometry.
5.1.2 Model Formulation
The governing equations for an unsteady, Newtonian two-phase system in domain
with boundary are given as follows
0, v (5.1)
2 T
st bPt
vv v v v v f f (5.2)
2
3 2 2
1 .
Mt
u r K
v ,
(5.3)
Where, t , , v , P , and , stand for time, effective density, velocity, pressure, and
effective dynamic viscosity, respectively. st f represents the surface tension
force and bf stands for the body force, such as the gravitational force g . The
mobility is denoted by ,M is the chemical potential and the parameters 1, ,K r u
describe the free energy F (Qian et al., 2003)
2 2 4
1
1 1 1.
2 2 4F K r u (5.4)
The interface thickness and the surface tension are 1/ ,K r and 2
12 2 / 3 ,r u
respectively. The phase field variable is defined as 1 / 1r u . 1
represents the phase with index i , and 1 the phase with index o . The effective
77
density and the effective viscosity of the two-phase system are considered as a
smooth function of the phase field variable
1 1,
2 2
1 1.
2 2
i o
i o
(5.5)
Since both the geometry and the expected solution are axisymmetric, the problem is
treated in axisymmetric coordinate system r r z zp p p i i with the basis vectors
; ,r z i and the coordinates ; ,p r z . The axisymmetric coordinates and basis
vectors are expressed by the three-dimensional Cartesian coordinate system
x x y y z zp p p p i i i with the basis vectors ; , ,x y z i and the coordinates
; , ,p x y z as
1/ /22 2 , , cos sin ;
arctan / , .
r x y z z r x y
y x z z
p p p p p p p
p p p
i i i
i i (5.6)
The axisymmetric form of the governing equations (5.1)-(5.3) for the mass
conservation, for the momentum in ri and zi directions, and for the phase-field
transport is:
0,r r z
r r z
v v v
p p p
(5.7)
2 2
2 2 2
12
,
r r r r r r r rr z
r z r r r r r z r r
r zr
z z r r
v v v v v v v vPv v
t p p p p p p p p p p
v vg
p p p p
(5.8)
2 2
2 2
1
2 ,
z z z z z zr z
r z z r r r z
r z zz
r z r z z z
v v v v v vPv v
t p p p p p p p
v v vg
p p p p p p
(5.9)
78
2 2
2 2
2 23
1 2 2
1,
1.
r z
r z r r r z
r r r z
v v Mt p p p p p p
u r Kp p p p
(5.10)
5.1.3 Initial and Boundary Conditions
The governing equations are solved subject to the following initial and boundary
conditions. Initially, both fluids are at rest and the inner fluid fills the inner tube as
described by the following profile
, m / s,
1 0 and 0
1 elsewhere
r z
r i z
p p
p R p H
v 0
(5.11)
For the velocity components, the no-slip boundary conditions are applied on the solid
wall of both inner and outer tube. On the symmetry line, symmetric conditions are
prescribed and at the outlet Neumann condition for the axial component of velocity
and Dirichlet condition for the radial component of the velocity is applied. On the
other hand, the boundary condition for phase field variable , are non-penetration
boundary conditions on the symmetry line, on the outlet and on the solid wall of the
inner and the outer tube.
0, 0,
0, 0
outlet outlet
outlet outlet
zr
vv
n
n n
(5.12)
0, 0,
0, 0,
symmetry symmetry
symmetry symmetry
rzv
v
n
n n
(5.13)
,,0 ,0p R p H p R p L
r i z r o z
v 0 v 0 (5.14)
,0 ,0
,0 ,0
0, 0,
0, 0,
p R p H p R p Lr i z r o z
p R p H p R p Lr i z r o z
n n
n n
(5.15)
79
For both inner and outer fluid, the Dirichlet conditions for velocity and phase field
variable at the inlet are as follows
2
, ,0 0
2
,0 0
2
0 0
1 0
2 1 , 0
or
2 1 , 0
r i
i
r i
i i
r
ir i r i
i
p R p Rr i
rz rp R p R
r i
z rp R p R
pQv v
R R
pv v v
R
(5.16)
2 2
*
*
*
*
1, 0,
2 , 0,
or
2 , 0,
i r o i r o
o
i r o i r o
o i
oi r o i r o
R p R R p R
z rR p R R p R
z rR p R R p R
Q av v
R R b
av v v
b
(5.17)
where,
2 2 2 2
2 2 2 2* *log .
log log
o i r o i
i r o i
io o
i i
R R p R Ra R p b R R
RR R
R R
(5.18)
where, ,i o
v v are the average velocity of inner and outer fluid, respectively.
5.2 Results and Discussions
5.2.1 Sensitivity Study of Node Density
Sensitivity study of the results in terms of meshless node-density is performed.
Dripping to jetting transition is analysed using inner (polydimethylsiloxane (PDMS)
oil) and outer (water) fluid velocities as 0.2 m / s, 1.6 m / si ov v . The material
properties for PDMS oil and water are given in Table 5.1. A related Non-dimensional
system, given in Appendix A, is used for simulations on three different node
arrangements, 30x400, 45x600 and 60x800 with required time steps 410 , 40.75x10
80
and 510 , respectively. In all cases, both Courant-Friedrichs-Lewy (CFL) and von
Neumann stability conditions are satisfied. The computational time per iteration on
coarse node arrangement (30x400) is 0.17 s and for the fine node arrangement
(60x800) is 1.79 s . The simulations are performed on an Intel Xeon processor
running at 2.0 GHz.
Table 5.1. Material properties used in simulations.
Material property Symbol Value
Viscosity of inner fluid i 0.01 kg/(m s)
Viscosity of outer fluid o 0.001 kg/(m s)
Density of inner fluid i 3970 kg/m
Density of outer fluid o 31000 kg/m
Surface tension 0.04 N/m
Mobility M 10 41.1785x10 m / Ns
Interface width 610 m
The dimensionless jet length L j , measured from the inner tube exit to the neck (see
Fig. 5.2), is used as an indication for convergence of the results.
Fig. 5.2. Illustration of the definition of the jet length L j and the limiting length
Ld.
81
Fig. 5.3. represents L j as a function of dimensionless time /i it tv R . It is observed
that the results obtained with the node arrangements 30x400, 45x600 and 60x800 do
not show a significant difference in the jet length. It is thus assumed that the results
with node arrangement 30x400 are reasonably accurate. Therefore, it is not required
to use finer node arrangements, which are very expensive in terms of computational
time, for further simulations.
Fig. 5.3. Dimensionless jet length L j as a function of dimensionless time t for the
different node arrangements.
5.2.2 Comparison with Finite Volume Results
In this subsection, the meshless results are compared with the results obtained with
an open source numerical toolbox OpenFOAM (Weller et al., 1998). The latter is
based on FVM (Patankar, 1980) discretization and VOF interface tracking method. It
has an ability to efficiently handle both structured and unstructured meshes. It uses
Piecewise Linear Interface Construction (PLIC) (Youngs, 1982) algorithm for
interface reconstruction and counter-gradient approach (Weller, 2008) to avoid the
interface smearing. Furthermore, it calculates the interface curvature using
Continuum Surface Force (CSF) (Brackbill et al., 1992) model. In order to solve the
partial differential equations PIMPLE algorithm is used. It is a combination of
Pressure Implicit with Splitting of Operators (PISO) (Issa, 1986) and Semi-implicit
82
Method for Pressure Linked Equations (SIMPLE) (Patankar and Spalding, 1972). It
also provides the adjustment of the time step during the simulation in order to ensure
the solution convergence, done by limiting the Courant number (Courant et al.,
1967).
For comparison of the results, the governing equations (5.1)-(5.5), subject to the
initial and boundary conditions given in Eqs. (5.11)-(5.18), with material properties
from Table 5.1 and an increased domain length 800 mL are considered. The
simulations are performed on 30x800 node arrangement with time step 62.27x10 ms .
Two different cases are simulated in order to analyse the dripping and jetting
phenomena. A comparison of the present results and FVM results calculated with a
similar mesh 30x800 is shown in Fig. 5.4 and Fig. 5.5, respectively. It is observed
(Fig. 5.4) that the dripping occurs at the low flow rates of both inner and outer fluid
set at 0.44 m / s, 0.3 m / si ov v . When the flow rate of outer fluid is increased to
0.9 m / sov , while keeping the same flow rate of the inner fluid, the drop size
decreases until the jet is formed and a breakup occurs downstream at the end of the
thin jet (Fig. 5.5). The meshless phase field simulations are in close agreement with
FVM-VOF results.
83
a
b
Fig. 5.4. A comparison of DAM and FVM results using
0.44 m / s, 0.3 m / si ov v at (a) 1 mst and (b) 2 ms.t
a
b
Fig. 5.5. A comparison of DAM and FVM results using
0.44 m / s, 0.9 m / si ov v at (a) 1 mst and (b) 2 ms.t
84
5.2.3 Sensitivity Study with Respect to the Process Parameters
This subsection studies the effect of dimensionless numbers such as capillary number
Ca /i iv and viscosity ratio /o i on two dimensionless parameters: the
limiting length Ld and the volume of the drop Vd . The limiting length Ld is
measured as a distance from the exit of the inner tube to the tip of the drop at breakup
(see Fig. 5.2) and the volume of the drop is calculated by using 2V ,d i i dv R t where
dt is the time needed to form a drop. For this purpose, the non-dimensional form of
governing equations ((A.1)-(A.6)) and the boundary conditions ((A.7)-(A.12)) is
considered. The time step is adjusted at the start of each simulation to satisfy the von
Neumann stability condition. The simulations are performed on 30x400 node
arrangement with time step 410 for Re 1,10 and Re 100 , however smaller time
step 510 is used for Re 0.01 . For all cases, the following dimensionless
numbers, Q 10,r and 2 Ca , are fixed.
5.2.3.1 Effects of the Capillary Number
The variation of limiting length Ld and volume of the drop Vd as a function of the
capillary number Ca (defined in Appendix A) is shown in Fig. 5.6 with the markers.
It is analyzed for three different density ratios 0.1,1.0 and 10 , while
keeping other dimensionless parameters fixed (e.g Re 0.01, Bo 0.01 and
D 0.05, 1.0c ). For small values of capillary number Ld first decreases and
then smoothly increases with increasing Ca . The curves of Ld for both 0.1 and
1.0 are nearly the same. On the other hand, Vd keeps decreasing as Ca
increases for all three cases. It is observed that for small Ca the surface tension force
is larger as compared to the viscous force, and longer time is needed for a drop to
pinch off and more fluid can flow into the drop, which increases the drop size (Fig.
5.8). On the other hand, as capillary number increases the viscous force becomes
dominant, causing the drop to move for longer distance with smaller size before the
breakup and also reduces the time period for drop formation (Fig. 5.8). The results
are found to be in close agreement with the reference results (Liu and Wang, 2015).
The limiting length Ld as a function of dimensionless time and interface profile for
85
Ca 0.01 at dimensionless time 87t and Ca 0.05 at 74t for 0.1 is
shown in Fig 5.7 and Fig. 5.8, respectively.
Fig. 5.6. Variation of limiting length Ld (left) and volume of the drop Vd (right)
as a function of capillary number. The solid lines represent the finite difference
results (Liu and Wang, 2015) and the markers show the results from this study.
Fig 5.7. Dimensionless limiting length Ld as a function of dimensionless time t
for (a) Ca 0.01 and (b) Ca 0.05 for 0.1 .
86
a
b
Fig. 5.8. The interface profile for (a) Ca 0.01 at 87t and (b) Ca 0.05 at
74t for 0.1 .
5.2.3.2 Effects of the Viscosity Ratio
The effect of viscosity ratio on limiting length Ld and volume of the drop Vd is
analysed for three different Reynolds numbers Re 1,10 and Re 100 . The other
dimensionless parameters are taken as Bo 0.01, Ca 0.01, D 0.05c and 0.1 .
The meshless results are plotted with the markers and the finite difference results
with the solid lines, shown in Fig. 5.9. Fig. 5.9(left) shows that Ld is continuously
decreasing as increases from 310 to 010 for Re 100 and for Re 1,10, it keeps
on decreasing when increases from 310 to 110 and then Ld increases for all three
cases. Similarly, Fig. 5.9 (right) shows that Vd also decreases with the increase of
. It is observed that for small values of Re and the limiting length is smaller.
The reason is that for smaller Re and , the inertial forces of inner fluid as well as
viscous drag forces of outer fluid are smaller as compared to the surface tension force,
forcing the drop to breakup near the orifice. As increases the viscous drag force
of outer fluid increases, which pushes the drop for a longer distance in downstream
and finally a jet is formed. The higher Re increases the inertial force of inner fluid,
which also pushes the drop for a longer distance. The results are found to be in close
87
agreement with the reference results (Liu and Wang, 2015). The limiting length Ld
as a function of dimensionless time for 1.0 and 2.0 at Re 100 is
plotted shown in Fig. 5.10, which shows significant change in Ld . Furthermore, the
interface profile for 1.0 at 82.9t and 2.0 at 106.9t for Re 100 is
shown in Fig. 5.11. It is observed that for 2.0 the limiting length is larger due
to the increasing viscous drag force of the outer fluid.
Fig. 5.9. Variation of limiting length Ld (left) and volume of the drop Vd (right)
as a function of viscosity ratio. The solid lines represent the finite difference results
(Liu and Wang, 2015) and the markers show the present results.
Fig. 5.10. Dimensionless limiting length Ld as a function of dimensionless time t
for (a) 1.0 and (b) 2.0 at Re 100 .
88
a
b
Fig. 5.11. The interface profile for (a) 1.0 at 82.9t and (b) 2.0 at
106.9t for Re 100 .
Equation Chapter (Next) Section 1
89
6. Conclusions
In this dissertation, an application of the combination of PFM and DAM for free and
moving boundary problems is studied. The proposed combination is applied for
solving RT instability and dripping-jetting phenomena. The obtained results
demonstrate that the blend of the two methods is effectively applicable to two-phase
problems with free and moving boundaries.
This chapter summarizes and concludes the work done in the present dissertation and
comments the continuation.
6.1 Summary of the Performed Work
The work performed in the framework of this dissertation can be summarized as
follow:
Physically different phase field models have been applied for solving two-
phase flow problems. The different models are as follows:
the Boussinesq phase field model for small density ratios with
constant viscosity and buoyancy force by neglecting surface tension
effect (Talat, Mavrič, Hatić, et al., 2018),
a variable density and constant viscosity model for small density
variations without surface tension effect and considering buoyancy
force (Talat, Mavrič, Hatić, et al., 2018),
a variable density model for large density variations considering
constant viscosity with an additional phase field dependent term in
90
momentum equation accounting for the surface tension effect (Talat,
Mavrič, Hatić, et al., 2018), and
a variable density and viscosity model with surface tension effect for
large density variations (Talat, Mavrič, Belšak, et al., 2018).
DAM was used for the spatial discretization of the governing equations that
describes the dynamics of two-phase flow. An incremental pressure
correction scheme was used for the pressure-velocity coupling.
The accuracy of the developed DAM discretization was studied by comparing
the meshless results with FVM and FDM results. For this purpose, the RT
instability problem in 2D and axisymmetric liquid jet problem was chosen.
The 2D RT instability problem served as an elaboration and verification for
the following numerical models parameters:
three different values of shape parameters 2.5, 5c and 10c ,
three different number of nodes in local subdomain i.e., 9, 11 and 13,
three different node arrangements 64x256, 128x512 and 192x786,
effect of different Atwood number At 0.01, 0.1, 0.3 and 0.5 on the
height of bubbles and spikes,
effect of the surface tension force on the dynamics of the interface.
The axisymmetric liquid jet problem was considered and analysed for the
verification of the following
jet length as a function of three different node arrangements 30x400,
45x600 and 60x800,
effect of the capillary numbers on the dimensionless limiting length
and volume of the drop,
effect of the viscosity ratios on the dimensionless limiting length and
volume of the drop.
6.2 Conclusions
The following conclusions can be drawn from the performed studies:
The combination of PFM and DAM was used for 2D benchmark RT
instability (dimensional form) and axisymmetric liquid jet problem (non-
91
dimensional form). It was found that the combination of PFM and DAM is
capable of solving free surface flows with large topological changes and
provides a valuable numerical tool for solving immiscible convective
hydrodynamics problems.
In RT instability problem, a sensitivity study of shape parameter in Gaussian
weight function was carried out. Three different values of shape parameter
2.5, 5.0c and 10c were used for the fixed number of 9 nodes in the local
subdomain. It was found that 10c is a most suitable value and produced the
results in close agreement with the FVM results obtained by the open source
code Gerris. The other two values produced results similar to each other, but
distinct from the FVM results. Since the selection of shape parameter is
problem dependent and there are no straightforward techniques to find a
suitable shape parameter for DAM, it was therefore suggested that results
should be evaluated for all three shape parameters to choose the optimum.
Afterwards, the sensitivity study of the number of the nodes in local
subdomain was performed by fixing shape parameter 10c . The number of
nodes in the local subdomain was taken 11 and 13, which produced the same
results and were in close agreement with FVM results (Popinet and Zaleski,
1999).
To study the node density convergence of the method, the solution was
compared for three different node arrangements 64x256, 128x512 and
192x786. It was found that there was a slight difference between the results of
node arrangement 64x256 and 128x512 but the results for node arrangements
128x512 and 192x786 were overlapping.
The effect of dimensionless Atwood number on the height of bubbles and
spikes using Model-II and Model-III (see Chap.4. ) was carried out. It was
observed that the symmetry of the heavier and the lighter fluids was
preserved for Model-III and was not affected by the Atwood number. The
results for At 0.1 were similar for both Model-II and Model-III but for
At 0.3 and At 0.5, the heavy front moved faster than the light front. It
was also found that the symmetry of the results was lost for Model-II. Model-
II and Model-III were considered to analyse the effect of different buoyancy
92
force on the dynamics of interface. The results were in close agreement with
the results obtained by staggered MAC method (Lee and Kim, 2012).
The dynamics of RT instability was also analysed with large density variation
and surface tension (Model-I). It was concluded that the shape of the
mushrooms was significantly affected by surface tension. The height of
bubbles and spikes was also increased due the large density variations, which
resulted into narrow trails and large curvature on both left and right tail of the
mushrooms. The meshless results were in excellent agreement with the FVM
results (Popinet and Zaleski, 1999).
For axisymmetric liquid jet problem, the dimensionless jet length (see Fig.
5.2) as a function of time was analysed for three different node arrangements
30x400, 45x600 and 60x800. It was found that there was no significant
change in the calculated jet length. The results with node arrangement 30x400
were reasonably accurate and used for further simulations.
Furthermore, the meshless results were compared with FVM-VOF results
obtained by OpenFOAM (Weller et al., 1998) in terms of dripping and jetting.
Dripping and jetting phenomena was analysed by changing the flow rate of
the outer fluid. The meshless results were in close agreement with the FVM
results in terms of drop size and temporal behaviour.
The effect of dimensionless capillary number in the range of 0.004 to 0.07 on
dimensionless limiting length Ld and volume of the drop Vd was analysed. It
was concluded that for small values of Ca longer time was required for a
drop to pinch off due to the large surface tension force. However, for higher
Ca the time period for drop formation was reduced and drop moved to the
longer distance before breakup due the high viscous force. The results were in
close agreement with the reference FDM results (Liu and Wang, 2015).
Similarly, the effect of the viscosity ratio on the dimensionless limiting
length Ld and volume of the drop Vd was analysed. It was observed that for
small value of , the limiting length was smaller due the small viscous force
of the outer fluid, which resulted in a drop breakup near the orifice. For
higher , the viscous drag force of outer fluid was higher, which pushed the
93
drop downstream for a longer distance and a jet was formed. The phase field
meshless simulations agree well with FDM results (Liu and Wang, 2015).
It is also concluded that the combination of PFM and DAM was suitable for
handling the axisymmetric forced-flow moving boundary two-phase flow
problems in co-flow microfluidics. However, there are some limitations of the
current state of the proposed numerical approach. It was not yet suitable to
numerically simulate the gas focused micro-jet in gas dynamics virtual nozzle
due to the complex sharp edges of the nozzle geometry and the very large
density difference of gas (helium) and water. So, a special treatment of the
sharp edges of the nozzle domain and a suitable modification of pressure
velocity coupling for large density difference of gas and water is needed in
the perspective.
6.3 Future Work
The phase field models for constant and variable density, viscosity and with and
without surface tension effect are used to analyze the dynamics of 2D RT instability
and axisymmetric liquid jet problems. The fluids are considered to be Newtonian,
incompressible and immiscible in the present research work. In the future, the long
time simulation for RT instability will be performed using DAM-PFM.
In the future, phase field formulation for compressible flow (Liu et al., 2016) of
mixing and non-mixing fluids will be numerically solved using DAM. A phase field
model will be developed for the numerical simulation of compressible gas phase and
incompressible liquid phase. Furthermore, numerical simulations will be performed
for mixing and non-mixing fluids in double flow focusing. The effect of
electromagnetic fields on liquid jet will be studied.
6.4 Publications
The performed research work in this dissertation resulted in the following
publications and presentations.
94
6.4.1 Journal Papers
Talat, N., Mavrič, B., Hatić, V., Bajt, S., and Šarler, B. (2018). Phase field simulation
of Rayleigh-Taylor instability with the meshless method. Engineering Analysis with
Boundary Elements. 87:78-89.
Talat, N., Mavrič, B., Belšak, G., Hatić, V., Bajt, S., and Šarler, B. (2018).
Development of the meshless phase field method for two-phase flow. International
Journal of Multiphase Flow. doi: 10.1016/j.ijmultiphaseflow.2018.06.003.
6.4.2 Conference Presentations
Šarler, B., Belšak, G., Talat, N., Zahoor, R., and Bajt, S. (2017). Modeling and
simulation of gas-focused micro jets = Modeliranje in simulacije plinsko fokusiranih
mikro curkov. The 16th Symposium of Physicists of the University of Maribor,
Maribor.
Šarler, B., Dobravec, T., Hatić, V., Hanoglu, U., Maček, M., Mavrič, B., Talat, N.,
and Vertnik, R. (2017). An overview on collocation meshless approach for solving
multiscale and multiphysics problems. International Conference on Computational &
Experimental Engineering and Science (ICCES), Funchal, Madeira Island, Portugal.
95
Equation Chapter (Next) Section 1
Appendix A Non-dimensional Form of the Governing
Equations
In order to reformulate the dimensional phase field model into dimensionless form,
the defined non-dimensional variables for space coordinates, velocity, density,
viscosity, pressure, time, phase field variable and chemical potential are as follows
/ , / , / , / ,
/ , / , / , / .
c c c c
c c c c c c c
l v
P Pl v t tv l
p p v v
(A.1)
The characteristic values are as follows
2
1, / , , , / , .c i c i i i c i c i c c cl R v v Q R r u r (A.2)
By inserting these variables into Eqs.(5.1)-(5.5), the obtained non-dimensional phase-
field model is as follows
0, v (A.3)
2Re B
Bo1 ,
Ca
T
z
Pt
vv v v v v
g (A.4)
1 1 1 1
,2 2 2 2
(A.5)
2
c
2 3
D
/ /
t
v , (A.6)
In the Eqs.(A.1)-(A.6), the non-dimensional parameters are Reynolds number
Re /i i i iv R , ratio of inertial to viscous force, capillary number Ca / ,i iv
ratio between viscous force and surface tension force. Bond number
2Bo / ,i iR g ratio of gravitational to surface tension force, B 3 / 2 2 i iv
96
proportional to capillary number and diffusion coefficient 2
cD 3 / 2 2 i iM v R .
Cahn number iR and / , /o i o i are density and viscosity ratio,
respectively. Similarly, the non-dimensional form of initial and boundary conditions
is as follows:
, ,0 0,
1 0 / and 0 /
1 elsewhere
i i
r z
r i z
p p
p R R p H R
v (A.7)
,/ ,0 / / ,0 /p R R p H R p R R p L R
r i i z i r o i z i
v 0 v 0 (A.8)
/ ,0 /
/ ,0 /
/ ,0 // ,0 /
0, 0,
0, 0
r o i z i
r i i z ir o i z i
p R R p H Rr i i z i p R R p L R
p R R p H Rp R R p L R
n n
n n
(A.9)
2
0 / / 0 /
0 /0 /
1, 0,
2 1 , 0,
r i i r i i
r r i ir i i
p R R p R R
z r p R Rp R Rv p v
(A.10)
/ / / /
**
/ / / /**
1, 0,
2Q , 0,
i i r o i i i r o i
i i r o i i i r o i
R R p R R R R p R R
z R R p R R r r R R p R R
av v
b
(A.11)
2 2
2 2**
2 2
2 2**
/ // log ,
log
/ // /
log
o i i i
i i r r
o
i
o i i i
o i i i
o
i
R R R Ra R R p p
R
R
R R R Rb R R R R
R
R
(A.12)
rQ /o iv v , ratio of average velocity of outer to inner fluid.
97
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