bib-pubdb1.desy.debib-pubdb1.desy.de/record/410389/files/PhD thesis.pdf · Acknowledgements First and foremost, I would like to express my profound gratitude to my mentor Prof. Dr.

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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