Development of an Immersed Boundary Method for the Simulation ofMoving Bodies at Fluid-Fluid Interfaces

by

Adam O’Brien

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial EngineeringUniversity of Toronto

c© Copyright 2019 by Adam O’Brien

Abstract

Development of an Immersed Boundary Method for the Simulation of Moving Bodies at

Fluid-Fluid Interfaces

Adam O’Brien

Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2019

Particles interacting with interfaces in multiphase flows appear in many scenarios of industrial

interest, including bitumen extraction, mineral recovery via froth flotation, soil remediation,

and many others. This thesis presents the development of Immersed Boundary Methods (IBMs)

coupled to a two-phase incompressible flow solver. The phases are treated as immiscible, and

are modelled using the Volume-of-Fluid (VOF) method. The methods developed to include the

effects of surface tension and contact lines are presented. A detailed description of the numerical

methods, as well as their implementation into a new software framework, are presented. Details

regarding the solution of the resulting large systems of sparse equations in parallel are briefly

described. The accuracy of the proposed models are demonstrated in three chapters; test cases

range from validation of the correct contact angle for droplets sitting on curved surfaces, to

more complex Fluid-Structure Interaction (FSI) problems involving moving objects at fluid-fluid

interfaces.

Stationary IBMs were coupled with a VOF multiphase solver, with the effects of surface

tension included via the Continuum Surface Force (CSF) method. The domain was separated

into fluid and solid regions, with coupling between them achieved through a ghost-cell interpo-

lation procedure, which in effect prescribed the desired boundary conditions for the velocity and

pressure fields. A new method was developed to include the effects of contact angle/wetting

for moving boundary problems. For these problems, a unified momentum equation for both

the fluid and solid phases was solved, and the effect of the boundary was introduced into the

momentum equation via a body force term. Solutions to FSI problems involving strong capil-

ii

lary forces are presented and compared to experiments. Finally, an application of the proposed

numerical methods to simulating porous media injection is shown. The algorithm’s ability to

predict an interface’s transition between stable displacement, viscous fingering and capillary

fingering is demonstrated.

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To my wonderful family, to whom I am forever indebted...

Acknowledgements

There are many people I have to thank, all of whom have supported me both academically

and/or personally throughout my studies. I’ve gained a great deal of both personal and technical

skills during my time at the University of Toronto, which I believe will serve me for the rest of

my life. In particular, I will always remember that:

1. You are not as smart as you think you are in your best moments, nor are you as dull as

you think you are in your worst. The truth is closer to the mean.

2. It’s not the compiler.

Firstly, I must thank Professor Markus Bussmann, my supervisor and the chair of the

Department of Mechanical and Industrial Engineering at the University of Toronto, for his

guidance, support and patience as I worked to complete this project. I’m very thankful for

both his technical expertise, as well as his positive and approachable attitude. Secondly, I must

also thank Professor Shahriar Afkhami of the New Jersey Institute of Technology, with whom

I had the privilege of working with closely during the later part of my studies. The advice and

knowledge he shared was paramount in bringing the last stages of this thesis to completion.

On the personal side, I must thank my parents Deb and Bob. Both of my parents have been

incredible supporters and mentors to both my sister and I. From an early age, they instilled

in me a love of learning and a curiosity about the world that persists today, without which I

would have never undertaken this journey. I’ve learned many important life lessons from them,

which were not always received with the gratitude they deserved at first. Their long list of

accomplishments in life have also helped form and inspire the goals I’ve set for myself. I’d also

like to thank my sister, whose many accomplishments are another source of inspiration.

I must also thank my partner Michelle, who has always supported me through highs and

lows for the last several years. She truly made my time in Toronto much more enjoyable, and

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always believed in what I was doing. Additionally, I’d like to thank some of my past lab mates.

I’d like to thank Pedro Isaza, whom I worked with on several other projects. His passion for

intellectual pursuits was second to none, and in working with him, I was pushed to learn a

great deal about fields outside of my own. I’d also like to thank experimentalist turned expert

Python programmer Sasan Mehrabian for just existing, and for his part in making the lab a

fun place to work.

Finally, I’d like the thank the countless academics and industry experts for their decades of

work in the field of Computational Fluid Dynamics (CFD). The many creative and ingenious

developments they have contributed to the field are truly remarkable, and it seems like new

and improved developments are always just around the corner.

v

Contents

List of Tables ix

List of Algorithms x

List of Figures xi

Nomenclature xvii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Numerical Modelling of Interfacial Flows . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Interface Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Computing Surface Tension in VOF Frameworks . . . . . . . . . . . . . . 5

1.3 Numerical Modelling of Contact Lines . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Immersed Boundary Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Continuous Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Discrete Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.6 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Governing Equations 18

2.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Non-Dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vi

3 Numerical Methodology 23

3.1 Discretization of the Momentum Equation . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Fractional-Step Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Evaluation of Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Chorin’s Projection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Helmholtz–Hodge Decomposition . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.2 Pressure Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 A Consistent CICSAM Scheme for Computing Mass and Momentum Flux . . . . 29

3.4 Computation of Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.1 Computation of the Interface Normals . . . . . . . . . . . . . . . . . . . . 31

3.4.2 Computation of Interface Curvature . . . . . . . . . . . . . . . . . . . . . 33

3.5 Implementation of the Immersed Boundaries . . . . . . . . . . . . . . . . . . . . 34

3.5.1 Ghost-Cell Direct Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5.2 Piecewise Quadratic Indirect Approach . . . . . . . . . . . . . . . . . . . 38

3.6 Coupling of IBMs and VOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7 Strategies for Solving the Discretized Equations . . . . . . . . . . . . . . . . . . . 40

3.7.1 Solution of the Linear Systems of Equations . . . . . . . . . . . . . . . . . 41

3.7.2 Parallel Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

References 49

4 A volume-of-fluid ghost-cell immersed boundary method for multiphase flows

with contact line dynamics 62

4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.2 Discretization of the Navier-Stokes Equations . . . . . . . . . . . . . . . . 67

4.3.3 Discretization of the VoF Equation and Momentum Flux . . . . . . . . . 69

4.3.4 Computing the Surface Tension Force . . . . . . . . . . . . . . . . . . . . 71

4.3.5 Ghost-Cell Immersed Boundary Method . . . . . . . . . . . . . . . . . . . 74

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4.3.6 Implementing Contact Line Dynamics on Ghost-Cell Immersed Boundaries 78

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.1 Imposing a Static Contact Angle on an IB Cylinder . . . . . . . . . . . . 80

4.4.2 Imposing a Contact Angle on a Poorly Mesh-Aligned Plane . . . . . . . . 83

4.4.3 Droplet Impact Onto a Cylinder . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.4 Droplet Penetration into a Porous Media . . . . . . . . . . . . . . . . . . 87

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

References 91

5 A Moving Immersed Boundary Method for Simulating Particle Interactions

at Fluid-Fluid Interfaces 98

5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.1 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.2 Solid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4.1 Discretization of the Fluid Equations . . . . . . . . . . . . . . . . . . . . . 103

5.4.2 Immersed Boundary Method . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4.3 Immersed Boundary/Volume-of-Fluid Contact Line Extension . . . . . . . 107

5.4.4 Immersed Boundary Wetting Force . . . . . . . . . . . . . . . . . . . . . . 109

5.4.5 Computation of Solid Body Forces . . . . . . . . . . . . . . . . . . . . . . 109

5.4.6 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5.1 In-Line Oscillation of a Cylinder . . . . . . . . . . . . . . . . . . . . . . . 112

5.5.2 Droplet on a Static Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5.3 Sinking Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.4 Impact of a Superhydrophobic Sphere . . . . . . . . . . . . . . . . . . . . 116

5.5.5 Formation of Cylindrical Granular Rafts . . . . . . . . . . . . . . . . . . . 117

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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References 126

6 Pore-scale direct numerical simulation of Haines jumps in a porous medium

model 131

6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3 Computational Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4.1 Haines Jump Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4.2 Displacement Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.4.3 Displacement Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

References 147

7 Conclusions 151

7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

References 155

ix

List of Tables

4.1 Non-dimensional parameters for droplet penetration into a porous media. . . . . 88

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List of Algorithms

3.1 BiCGSTAB algorithm for solving Ax = b. . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 V-Cycle multi-grid algorithm for solving Ax = b. . . . . . . . . . . . . . . . . . . . 45

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List of Figures

1.1 Two phases represented as a VOF field on a cartesian grid. The numbers denote

the volume fraction of the cell occupied by the blue phase. . . . . . . . . . . . . . 4

1.2 A droplet with contact angle θe at equilibrium. . . . . . . . . . . . . . . . . . . . 6

1.3 Kernel used to transfer F to neighbouring fluid nodes. The shaded region depicts

the area of influence of the spreading function d. . . . . . . . . . . . . . . . . . . 9

1.4 The cell classifications for a ghost-cell IBM. . . . . . . . . . . . . . . . . . . . . . 11

1.5 Representation of the bilinear interpolation stencil used in the ghost-cell method.

IP denotes the image point, where values of φ are interpolated from the sur-

rounding nodes, while GC denotes the ghost-cell, where the prescribed value will

depend on the type of boundary condition. . . . . . . . . . . . . . . . . . . . . . 12

1.6 Representation of a cut-cell IBM, where cells adjacent to the body are truncated,

and FVM discretization is performed only on the portion of the cell within the

fluid. Yellow cells are regular cut-cells, red cells are small cut-cells, and grey cells

are solid cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 Schematic showing a generic body defining an IB domain, occupying volume Ωib

with boundary Γib. The cells are flagged using a common flagging scheme in

many DF IBMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Schematic of a typical FVM cell on which discretized Navier-Stokes equations

are solved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Stencil used for the least-squares reconstruction of the interface in the CELESTE

scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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3.3 Interpolation stencil used in the ghost-cell immersed boundary method, showing

the image point xip and boundary point xbp. . . . . . . . . . . . . . . . . . . . . 35

3.4 Least-squares stencil (red) constructed by expanding a biquadratic polynomial

to neighbouring fluid and forcing points. Dark gray cells are solid cells, light

gray cells are forcing cells. The dashed blue lines denote neighbouring stencils. . 39

3.5 Graphical depiction of the Schwarz preconditioner with an overlap of one. The

shaded region represents the local block. . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Illustration of the v-cycle multi-grid coarse grid levels. Rk denotes the applicable

restriction operator and Pk is the prolongation operator. . . . . . . . . . . . . . . 46

3.7 The partitions generated by the domain decomposition scheme. . . . . . . . . . . 47

4.1 Least-squares stencil used for both the evaluation of m and κ, for (a) a regular

cell and (b) boundary cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Interpolation stencil used in the ghost-cell immersed boundary method, showing

the image point xip and boundary point xbp. . . . . . . . . . . . . . . . . . . . . 75

4.3 Degenerate case resulting in an ambiguous choice of interpolation stencil. The

red lines show the two possible stencils based on a nearest intersection, while the

black stencil is the actual stencil used. . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Least-squares stencil used for the evaluation of κ near the IB surface. . . . . . . . 79

4.5 Initial configuration for a planar droplet on a horizontal cylinder. . . . . . . . . . 80

4.6 Static contact angles imposed on a cylindrical IB. . . . . . . . . . . . . . . . . . . 82

4.7 Numerically computed droplet radii at various contact angles. . . . . . . . . . . . 82

4.8 Mass loss error per unit non-dimensional time for the planar droplet situated on

a cylinder with θ = 30. Note that for this case, t∗max = 0.2. . . . . . . . . . . . . 83

4.9 Convergence of the proposed method with grid refinement, with θ = 30. . . . . . 84

4.10 Planar droplets situated on a flat, well-aligned IB. The bottom droplets are

placed on a regular domain boundary for reference. . . . . . . . . . . . . . . . . . 84

4.11 Planar droplets situated on a tilted, poorly-aligned IB. The bottom droplets are

placed on a regular domain boundary for reference. . . . . . . . . . . . . . . . . . 85

4.12 Internal droplet pressure at varying contact angles. . . . . . . . . . . . . . . . . . 85

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4.13 Non-dimensional droplet radius with respect to time for a planar droplet on the

flat platform, with θ = 30 for various Ohnesorge numbers. . . . . . . . . . . . . 86

4.14 Planar droplet impact onto a hydrophilic (θ = 30) cylinder. Frames correspond

to t = 0 s, t = 0.05 s, t = 0.07 s, t = 0.09 s and t = 0.2 s. . . . . . . . . . . . . . . 87

4.15 Planar droplet impact onto a hydrophobic (θ = 150) cylinder, at t = 0 s, t = 0.05

s, t = 0.07 s, t = 0.1 s and t = 0.2 s. . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.16 Droplet penetration into porous media case. . . . . . . . . . . . . . . . . . . . . . 88

4.17 Penetration of a planar droplet into multiple cylinders of varying wettability;

snapshots at t∗ = 5, t∗ = 8, t∗ = 10, t∗ = 15, t∗ = 20 and t∗ = 30. . . . . . . . . . 89

5.1 Least-squares stencil ( ) and neighbouring least-squares stencils ( ) constructed

by expanding a bi-quadratic polynomial to neighbouring fluid and forcing points.

Dark gray cells are solid cells, light gray cells are forcing cells. . . . . . . . . . . . 106

5.2 Contact line interpolation stencils for extending the γ field into the boundary. . . 108

5.3 Result of the contact line extension near an IB for (a) θ = 45 and (b) θ = 135.

The extension is performed for all cells between the extension boundary ( ) and

IB surface ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4 Interpolation stencil for the sharp interface capillary force calculation. . . . . . . 111

5.5 Time history of the drag coefficient for ∆h = 0.04D ( ), ∆h = 0.02D ( ),

∆h = 0.01D ( ), compared against the experimental data (). . . . . . . . . . . 113

5.6 Pressure contours (left) and vorticity magnitude countours (right) for the inline

oscillating cylinder, with KC = 5 and Re = 100. . . . . . . . . . . . . . . . . . . 119

5.7 Final configurations of the spherical droplets situated on solid spheres, with (a)

θ = 45, (b) θ = 90 and (c) θ = 135, and compared against the exact solution

( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.8 (a) Internal droplet pressure compared to the exact solution, and (b) droplet

radii compared to the exact solution. . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.9 Numerical setup for the horizontal sinking cylinder. . . . . . . . . . . . . . . . . . 121

5.10 Snapshots comparing the numerically simulated and experimentally determined

positions of a sinking horizontal cylinder. . . . . . . . . . . . . . . . . . . . . . . 121

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5.11 Comparison of numerical results ( ) against the analytical solution with capil-

lary forces ( ), analytical solution without capillary forces ( ) and experimental

data (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.12 Numerical set-up for the axisymmetric superhydrophobic sphere impact. . . . . . 122

5.13 Numerical ( ) and experimental () non-dimensional heights for (a) the sinking

sphere and (b) the bouncing sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.14 Snapshots of the sinking sphere simulation. . . . . . . . . . . . . . . . . . . . . . 123

5.15 Snapshots of the bouncing sphere simulation. . . . . . . . . . . . . . . . . . . . . 124

5.16 Set-up for the cylindrical granular raft simulation. . . . . . . . . . . . . . . . . . 124

5.17 Snapshots of the cylinder raft formation. . . . . . . . . . . . . . . . . . . . . . . . 125

6.1 Domain set-up for the Haines jump simulations. . . . . . . . . . . . . . . . . . . . 135

6.2 Simulation results displaying the Haines jump mechanism for θ = 30 ( ), θ =

90 ( ), and θ = 150 ( ) for Ca = 1× 10−3. . . . . . . . . . . . . . . . . . . . 138

6.3 Simulation results displaying the Haines jump mechanism for θ = 30 ( ), θ =

90 ( ), and θ = 150 ( ) for Ca = 2.5× 10−3. . . . . . . . . . . . . . . . . . . 139

6.4 Simulation results displaying the Haines jump mechanism for θ = 30 ( ), θ =

90 ( ), and θ = 150 ( ) for Ca = 5× 10−3. . . . . . . . . . . . . . . . . . . . 140

6.5 Displacement patterns at varying Ca and θ. Above a certain Ca threshold only

viscous fingering is visible for the parameters studied in this work. . . . . . . . . 141

6.6 Injection simulation for θ = 30, with Re = 5 and Ca = 1× 10−3. . . . . . . . . 142

6.7 Injection simulation for θ = 90, with Re = 5 and Ca = 1× 10−3. . . . . . . . . 142

6.8 Injection simulation for θ = 150, with Re = 5 and Ca = 1× 10−3. . . . . . . . . 143

6.9 Driving pressure (∆p∗ = ∆pDµ|U| ) across the porous media domain, for (a) Ca =

10−1, (b) Ca = 10−2 and (c) Ca = 10−3, for θ = 30 ( ), θ = 90 ( ) and

θ = 150 ( ). Interface mergers/pore-filling events can be identified by rapid

changes in the pressure signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

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6.10 Comparison of front location of the invading fluid in the longitudinal direction,

for (a) θ = 30, (b) θ = 90 and (c) θ = 150, with Ca = 10−1 ( ), Ca = 10−2

( ) and Ca = 10−3 ( ). As the capillary number decreases, the movement of

the front becomes increasingly dominated by capillary rise between pores and

Haines jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.11 Displacement fraction S of the invading fluid with respect to the pore volume,

for (a) Ca = 10−1, (b) Ca = 10−2 and (c) Ca = 10−3, with θ = 30 ( ), θ = 90

( ) and θ = 150 ( ). In (a), saturation takes longer due to the presence of

capillary fingering and the formation of droplets. The kinks in (b) and (c) are

due to residual fluid pockets which are not displaced. . . . . . . . . . . . . . . . . 145

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Nomenclature

Acronyms

AMG Algebraic Multi-Grid

BiCGSTAB Bi-Conjugate Gradient STABilized

CF Continuous Forcing

CFD Computational Fluid Dynamics

CGS Conjugate Gradient Stabilized

CICSAM Compressive Interface Capturing Scheme for Arbitrary Meshes

CSF Continuum Surface Force

DF Discrete Forcing

FVM Finite-Volume Method

GMRES Generalized Minimum RESidual

HPC High-Performance Computing

HRIC High Resolution Interface Capturing

IBM Immersed Boundary Method

ILU Incomplete Lower-Upper (factorization)

MAC Marker-and-Cell

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MOF Moment-of-Fluid

MPI Message Passing Interface

PDE Partial Differential Equation

PLIC Piecewise Linear Interface Calculation

SFO Spurious Force Oscillation

SLIC Simple Line Interface Calculation

SOR Successive Over-Relaxation

TFQMR Transpose-Free Quasi-Minimal Residual

VOF Volume-of-Fluid

Non-Dimensional Numbers

Bo Bond number

Ca Capillary number

Eo Eotvos number

KC Keulegan-Carpenter number

Oh Ohnesorge number

Re Reynolds number

We Weber number

Symbols

∆ Difference operator

δ Dirac delta function

fg Gravitational body force

xviii

fσ Surface tension body force

g Gravitational acceleration

Γ Domain boundary

γ Volume fraction

∇ Del operator

κ Interface curvature

µ Fluid viscosity

Ω Domain

Ωc Cell volume

ρ Fluid density

σ Surface tension coefficient

t Interface tangent unit vector

τ viscous stress tensor

θ Contact angle

γ Convolved volume fraction

F Force on a Lagrangian point

f Force on an Eulerian point

I Identity matrix

K Preconditioning matrix

sf Face outward normal

U Velocity in the Lagrangian reference frame

xix

X Position vector in the Lagrangian reference frame

x Position vector in the Eulerian reference frame

u Fluid velocity

n Interface normal unit vector

nw Wall unit normal

x Cartesian basis vector

C Colour function

D Eulerian to Lagrangian interpolation operator

d Spreading function

h Interface thickness

p Pressure

t Time

V Volume

xx

Chapter 1

Introduction

1.1 Motivation

Interfacial flows involving two or more immiscible fluid phases and dispersed particles appear

in many flows of practical interest. The understanding of the dynamics of such flows is crucial

to the continued development of many processes including the recovery of oil from underground

reservoirs [1], the remediation of contaminated soil [2], recovery of minerals through froth

flotation [3], inkjet printing [4], nano- and micro-particle drug delivery [5], alternative surfac-

tants [6], and many others. In order to understand such phenomena, the complex interplay

between the interfacial surface tension that exists between solid and liquid phases, hydrody-

namics, solid-body dynamics, as well as any external forces such as gravity must be considered.

Such interactions are highly non-linear, and therefore analytical approaches are suited only for

simplified, highly controlled model cases. In order to gain insight into more complex dynamic

systems, one must turn to empirical, semi-empirical or numerical modelling. Computational

Fluid Dynamics (CFD) is a natural choice of tool for modelling these systems, as it allows for

the solution of much more sophisticated governing equations.

Multiphase systems involving two or more fluids have been studied extensively using nu-

merical approaches, with significant progress having been made in modelling bubbles [7–11],

droplets [12–15], liquid jets [16–18] and other phenomena where surface tension plays a sig-

nificant role in the dynamics of the system. Significant attention has also been given to the

modelling of contact lines, when a fluid-fluid interface interacts with a solid [19–23]. Contact

1

line problems are inherently complex, involving highly disparate spatial scales, making their

modelling difficult and computationally intensive.

While substantial progress has been made in the modelling of fluid-fluid interfaces and their

interactions with stationary solid boundaries, only a handful of attempts have been made to

model the interaction of dynamic moving bodies at fluid-fluid interfaces. These are examples

of Fluid-Structure Interaction (FSI) problems with an additional fluid-solid coupling through

capillary forces. Since most problems of interest are likely to exist on the particle scale and

contain multiple particles with arbitrary relative motion, the usage of conventional body-fitted

meshes is impractical. An alternative is to forego the use of body-fitted meshes altogether

through the use of the so-called Immersed Boundary Method (IBM), where the influence of

solid boundaries is introduced into the governing equations through a forcing term rather than

as a boundary condition. This makes it feasible to model the interactions of multiple dynamic

bodies without the need for any re-meshing schemes.

This thesis consists of strategies for modelling particles, represented as solid bodies undergo-

ing rigid body motion, interacting with fluid-fluid interfaces using IBMs. The remainder of this

chapter is structured as follows; in Section 1.2, past numerical work concerning the modelling

of interfacial flows is discussed. In Section 1.3, numerical approaches to treating contact lines

on solid surfaces is overviewed. In Section 1.4, an overview of IBMs is presented. In Sections

1.5 and 1.6, the main objectives of the thesis, and its organization, are outlined.

1.2 Numerical Modelling of Interfacial Flows

The numerical modelling of interfacial flows in the context of this work refers to flows involving

two immiscible fluid phases, and the forces imparted on the fluid through the imbalance of

free surface energy at the interface. From a numerical standpoint, there are two primary

considerations when modelling such flows; the tracking of each phase as the flow field evolves,

and the computation of the surface tension force from the discrete description of the interface.

For an overview of various ways to represent an interface in a numerical model, as well as how

to select a method of interface advection and surface tension computation, see the review by

Scardovelli and Zaleski [9].

2

1.2.1 Interface Tracking

The first consideration when modelling interfacial flows is how to appropriately evolve the

interface with the local flow field. The various methods for representing an interface within

a simulation generally fall under two categories. The first category includes surface-tracking

methods such as the Marker-and-Cell (MAC) method [24] and front-tracking methods [25–

27], where the interface is represented by a discrete surface and is evolved in the Lagrangian

reference frame. The second category includes phase-tracking methods [28] such as the level-set

method [29, 30] and Volume-of-Fluid (VOF) [31] methods.

These two broad categories of approaches have distinct strengths and weaknesses. In surface-

tracking methods, a topological description of the interface is readily available, making the

computation of surface tension comparably easier; however, in cases where interfaces are ex-

pected to merge or break-up, specialized algorithms are needed to join/separate the discrete

Lagrangian surfaces. Additionally, exact mass conservation is generally not guaranteed by such

methods. On the other hand, with phase-tracking approaches, no additional treatment is needed

to handle merging and break-up, and mass conservation is guaranteed by some approaches such

as VOF [31] and Coupled Level-Set VOF (CLSVOF) [18]. However, a topological description

of the interface surface may not be readily avaiable, and must be obtained by reconstructing

the interface from phase field information at every time step in order to obtain the geometric

information needed to compute the surface tension. This process can be expensive, and also

leads to spurious force oscillations arising from numerical errors in the curvature computation

and force balance, which will be discussed further in Section 1.2.2.

Phase-tracking methods consist of one or more colour functions C which dictate how much

of a particular phase is contained within a cell, as represented by a distance from the interface

in the case of a level-set method, or the fraction of cell volume occupied by a particular phase

in a VOF method. As the flow field evolves, C is updated through the advection equation,

∂C

∂t+∇ · uC = 0 (1.1)

In the VOF method, the colour function C represents the fraction of the total cell volume,

with γ ∈ [0, 1], which is occupied by a particular phase. Evidently, when γ is either 0 or 1, the

3

0

0

0

0

0

0

0

0 0

0

0

0

0

0 0 0 0 0

Figure 1.1: Two phases represented as a VOF field on a cartesian grid. The numbers denotethe volume fraction of the cell occupied by the blue phase.

cell is immersed entirely within one of the phases, while an intermediate value indicates that the

cell is intersected by an interface. Fig. 1.1 depicts an interface projected onto a cartesian grid.

In a typical Finite-Volume Method (FVM), Eq. (1.1) would be discretized and solved using

some variation of the upwind FVM scheme, which is necessary to preserve the monotonicity,

and hence stability, of the solution. The issue with this approach is that standard upwind

schemes introduce unacceptable levels of numerical diffusion into the solution, which leads to

unphysical results when simulating interfacial flows where the separate phases are supposed to

be immiscible. Thus, specialized advection schemes are needed in order to minimize numerical

diffusion and retain a sharp, discontinuous interface between the phases.

Considerable effort has been invested in devising advection schemes for VOF methods, sev-

eral of which are overviewed in [32]. Some widely used schemes include geometric approaches

such as Simple Line Interface Calculation (SLIC) [31], Piecewise Linear Interface Calculation

(PLIC) [33–37], and the Moment-of-Fluid (MOF) method [38]. Geometric methods use a re-

constructed interface surface, such as a piecewise linear function in PLIC, and translate the

resulting polygons using the local flow velocities. This approach is highly successful in min-

4

imizing numerical diffusion, but can often make even an explicit advection scheme relatively

expensive, especially on irregular meshes. Alternatively, compressive high resolution differenc-

ing schemes can also be used, which combine upwinding and downwinding in a manner that

satisfies local boundedness to ensure stability [35], which are sometimes referred to as alge-

braic interface advection schemes. Such methods are comparable to geometric methods in their

ability to maintain a sharp interface, but are generally less expensive and can also be straight-

forwardly adapted to irregular meshes [39]. Examples include the High Resolution Interface

Capturing (HRIC) scheme [40] and the Compressive Interface Capturing Scheme for Arbitrary

Meshes (CICSAM) [41].

1.2.2 Computing Surface Tension in VOF Frameworks

The second component of numerically modelling interfacial flows involves the computation of

the surface tension forces from a description of the interface. In the context of VOF, this can

be particularly challenging since only a discrete Eulerian description of the phases is available.

The VOF field must therefore be used to reconstruct the surface topology of the interface before

the surface tension can be computed; in particular the interface curvature κ and interface unit

normal vector n. As noted by numerous authors, this technique leads to the generation of so-

called “spurious” or “parasitic” currents, which are unphysical velocities along the interface that

originate from errors in the surface tension computation [9, 32, 42–47], as well inconsistencies

between the discretization of surface tension and other body forces [48]. An explanation of

how a VOF method can lead to spurious currents is given by Harvie et al. [49]. In essence, an

actual VOF implementation differs from an ideal one in that when discretized, it represents the

averaged volume fraction over the dimensions of a discrete computational cell, resulting in sharp

directional perturbations in the computed n. This also manifests itself in directional variations

in κ, as well as other fluid properties. These erroneous forces are not curl-free, and thus cannot

be balanced by the irrotational pressure gradient term alone, so must instead be balanced by

one or more of the three other velocity dependent terms (inertial, advective and viscous), which

can only be non-zero if the velocity field itself is non-zero. For a recent comprehensive review of

several techniques of computing surface tension in VOF and level-set frameworks, see Popinet

[50].

5

The most famous and widely used method of computing surface tension in phase-tracking

frameworks is the Continuum Surface Force (CSF) method of Brackbill et al. [51]. In this

approach, n is defined by the gradient of the colour function, and the surface tension itself is

evaluated as a volumetric body force (see Section 3.4 for further details). It has been reported

previously by Renardy and Renardy [52] and Francois et al. [48] that using a consistent method

of computing gradients of the colour function, in conjuntion with using face-to-cell interpolation

of the body forces, greatly reduces spurious currents, particularly when κ is prescribed. Thus,

most solvers which resolve surface tension dominated flows make use of so-called balanced-force

algorithms.

1.3 Numerical Modelling of Contact Lines

A contact line exists whenever an interface between two fluids intersects a solid surface. Fig. 1.2

depicts a sessile droplet on a surface at equilibrium. The equilibrium contact angle θe forms as

a result of an imbalance between the interfacial energy of the solid-gas interface σsg, solid-liquid

interface σsl and liquid-gas interface σlg.

liquidgas

solid

θe

σlg

σsl σsg

Figure 1.2: A droplet with contact angle θe at equilibrium.

A simple balance of the interfacial forces depicted in Fig. 1.2 leads to the well-known Young’s

equation, Eq. (1.2):

6

cos θe =σsg − σsl

σsl(1.2)

For a droplet in equilibrium, θe is a thermodynamic property of the system which reflects the

difference between solid-liquid and solid-gas interfacial energies. Experiments have shown that

θe depends only on the properties of the solid material and of the liquid. The characterization

of moving contact lines is much more difficult, since in general there is a difference between the

advancing contact angle, θa and the receding contact angle, θr. This phenomenon is generally

referred to as contact angle hysteresis [53].

A number of authors have noted the well-known contradiction that exists when modeling

a moving contact line on a no-slip surface [54]. While the use of the FVM method creates an

implicit slip length which partially alleviates this singularity, the stresses themselves scale with

mesh refinement. Afkhami et al. [55] proposed a mesh-dependent contact angle model which

computes a slip length, and is based on the theoretical work of Cox [56]. In this approach, a

numerical contact angle θnum is determined which is dependent on the capillary number Ca,

and decouples the stress at the contact line from the size of the mesh. An overview of several

computational methods for handling moving contact lines was compiled by Sui et al. [22].

1.4 Immersed Boundary Methods

In lieu of using boundary conditions prescribed at the edges of body-fitted meshes to represent

the presence of solid bodies in CFD simulations, Immersed Boundary Methods (IBMs) can be

used to represent the effect of solid bodies in a flow by introducing a forcing term into the

Navier-Stokes momentum equation (it should be noted that IBMs can also be used to simulate

fluid-fluid and liquid-gas interfaces [9, 23]). Many approaches to computing the forcing exist;

some which compute the forcing term as a volumetric body force (generally referred to as

indirect forcing), and others which modify the advection and viscous operators directly (referred

to as direct forcing). The origin of the IBM is attributed to Peskin [57] who originally developed

the method to simulate the flow of blood through a heart, where the heart itself was modelled

as an infinitely flexible elastic membrane overlayed on a fixed cartesian grid. IBMs are generally

employed in scenarios where the bodies undergo high degrees of deformation and/or extensive

7

motion relative to other bodies with few or no constraints. Such scenarios make deforming mesh

approaches impractical, since frequent expensive re-meshing steps would be required. Today,

IBMs can be broadly categorized into so-called Continuous Forcing (CF) and Discrete Forcing

(DF) methods. An extensive review of some of the more popular IBM approaches can be found

in [58].

1.4.1 Continuous Forcing

The original IBM work of Peskin [57, 59, 60] is considered an example of a CF method. In a CF

method, the forcing term in the momentum equation is computed prior to discretization. This

usually means the body itself be treated as deformable and elastic, so that the feedback forces

on the surrounding fluid can be computed from the displacement of the Lagrangian markers

placed along the body, through a constitutive relationship such as Hooke’s Law. The body

itself is discretized into discrete deformable segments, the vertices of which are massless points

tracked in a Lagrangian fashion according to the equation,

∂Xk

∂t= u(Xk, t) (1.3)

where Xk is the position vector of the kth Lagrangian point. The stress on the segment is

imparted on the surrounding fluid through Newton’s second law, which takes the form,

f(x, t) =∑

Fk(t)δ(|x−Xk|) (1.4)

where f is a force imparted on the fluid, Fk is the force due to deformation on a Lagrangian

marker point, x is a position in the Eulerian reference frame, and δ is the Dirac delta function.

Since X does not typically coincide with a point on the grid such as a cell-centre, and the desire

is to compute the force the IB imparts on the fluid at computational points, δ in Eq. (1.4) is

replaced with an operator, d, which spreads the forcing on the Lagrangian points to a band of

cartesian grid cells surrounding the boundary. This spreading operator takes a form similar to

Eq. (1.4),

f(x, t) =∑

Fk(t)d(|x−Xk|) (1.5)

8

A graphical depiction of the spreading of Fk from a Lagrangian point to an Eulerian region

is shown in Fig. 1.3. Due to the same misalignment between the Lagrangian points and the

fluid nodes, the velocity of the Lagrangian points in Eq. (1.3) is interpolated from the Eulerian

grid points using the same kernel in Fig. 1.3.

F

k

Figure 1.3: Kernel used to transfer F to neighbouring fluid nodes. The shaded region depictsthe area of influence of the spreading function d.

The choice of spreading operator d in Eq. (1.5) is not obvious, and considerable attention in

the literature has been dedicated to identifying the best option. Beyer and Leveque [61] showed

that d can be chosen such that the spatial accuracy of the numerical scheme is preserved. A

formally second-order kernel is also presented by Lai and Peskin in [62], and more recently Bao

et al. [63] proposed a kernel with improved translational invariance compared to other common

force spreading kernels.

The drawback of the CF approach is that treating the body as rigid becomes difficult.

Increasing the stiffness of the body to a large value can lead to excessive time step constraints

in order to avoid poorly posed problems. Another CF approach involves treating the solid

boundary like a porous medium, governed by Navier-Stokes/Brinkman equations. The method

was proposed by Angot et al. [64] and Khadra et al. [65], and is known as a fictitious domain or

penalization method. While such methods are capable of simulating IBs that are rigid, they rely

9

on a permeability parameter, which results in an error in enforcing the boundary condition.

Similar to the method of Peskin, reducing the permeability too much results in an ill-posed

problem.

1.4.2 Discrete Forcing

The second major category of IB methods are DF methods. Unlike CF methods, the forcing

is applied after the governing equations have been discretized. DF methods require no special

treatment in order to impose rigid boundaries. As suggested by Mittal and Iaccarino [58],

DF methods can be further categorized in two ways; direct forcing where the BCs are applied

directly, and indirect forcing where the effect of the body is applied through a body-forcing

term in the Navier-Stokes equations, in a manner similar to the CF IBM method of Peskin [57]

described in the previous section.

Direct Forcing

In a direct forcing method, the operators for the advective and viscous terms are altered directly

to impose the boundary condition directly. This method has two primary advantages. The first

advantage of this approach is that the effect of the IB is highly localized, and limited to cells that

are directly adjacent to a cell embedded within the boundary. This is particularly advantageous

for moderate to high Re flows, where excessive force spreading might otherwise result in an

artifically large boundary layer. IBMs which localize the forcing to cells only in the immediate

vicinity of the IB are sometimes referred to as ‘sharp’ interface IBMs for this reason. Secondly,

the method itself does not affect the local accuracy of the discretization for cells near the IB,

thus no special treatment is required as in the case of the CF approach. Indeed, it can be

shown that the overall accuracy of a direct forcing IBM is preserved so long as interpolation

polynomials of appropriate order are chosen [66–68].

One of the most popular and simplest methods of direct forcing are so-called ghost-cell

methods. In the ghost-cell method, cells within the IB that neighbour a cell within the fluid

are flagged as ghost-cells, as shown in Fig. 1.4. The values of an arbitrary field φ defined on the

ghost-cells have their values replaced by an interpolated value, so that when φ is interpolated to

the surface of the IB, the correct boundary value of φ is recovered. For example, the ghost-cell

10

velocities are computed such that the no-slip condition is satisfied when interpolating to the

IB surface. This results in a feedback force from the advective and viscous operators in the

momentum equation, although the value of this force need not be computed explicitly.

Fluid cell Solid cell Ghost cell

Ωib

Γib

Figure 1.4: The cell classifications for a ghost-cell IBM.

The type of interpolation must also be carefully chosen. A popular choice of interpolation,

originally proposed by Majumdar et al. [69], is a bilinear interpolation, which takes the form,

φ(x, y) = a1xy + a2x+ a3y + a4 (1.6)

where x and y denote the spatial coordinates at an image point (see Fig. 1.5), and a1, a2,

a3, a4 are the unknown interpolation coefficients which must be determined from the desired

boundary condition type and the four cells which surround the image point. Many other choices

of interpolation also exist; for example, an interpolation that is quadratic with respect to the

wall normal direction and linear with respect to the wall tangent direction was proposed by

Majumdar et al. [69],

φ(n, t) = a1n2 + a2nt+ a3n+ a4t+ a5 (1.7)

11

while a bi-quadratic interpolation was similarly proposed by Tseng and Ferziger [68]. It was

concluded that for standard second-order FVM schemes, no appreciable difference exists when

using higher order interpolation schemes. Regardless of the chosen interpolation scheme and

boundary condition type, a set of equations of the form,

∑αiφi = φgc (1.8)

is formed, where φi is the ith cell-centred value of φ used in the interpolation, αi is its interpola-

tion weight, and φgc is the interpolated ghost-cell value. The explanation of how the coefficients

are computed for various boundary condition types is deferred until Section 3.5.1.

1 2

3 4

IP

GC

Figure 1.5: Representation of the bilinear interpolation stencil used in the ghost-cell method. IPdenotes the image point, where values of φ are interpolated from the surrounding nodes, whileGC denotes the ghost-cell, where the prescribed value will depend on the type of boundarycondition.

Ghost-cell methods have been used successfully for a wide range of both static and dynamic

boundary problems, including swimming fish and insect flight [70], particle-laden flows [71], and

vortex-induced vibration of a cylinder [72]. One potential drawback of the ghost-cell method

is the generation of Spurious Force Oscillations (SFOs) during moving boundary problems [73,

74]. Seo and Mittal [73] identified two primary culprits in the generation of SFOs; errors in the

so-called ‘geometric conservation law’, which are essentially changes in the represented volume

of the body on the Eulerian grid as the classification of cells change from fluid to ghost-cells

12

Figure 1.6: Representation of a cut-cell IBM, where cells adjacent to the body are truncated,and FVM discretization is performed only on the portion of the cell within the fluid. Yellowcells are regular cut-cells, red cells are small cut-cells, and grey cells are solid cells.

and vice versa, and local changes in the effective geometry of the IB. Seo and Mittal [73]

proposed a hybrid ghost-cell/cut-cell method and a virtual merging technique to address the

mass conservation issues leading to SFOs, which was effective in reducing SFOs by roughly an

order of magnitude.

In the ghost-cell method, strict enforcement of momentum and mass conservation is not

possible. The only way in which these can be guaranteed is through a pure FVM approach,

which is the motivation behind cut-cell IBMs. The original cut-cell method was put forth by

Clarke et al. [75] for solving the Euler equations around airfoils, and was later adapted to

viscous flows [66, 67]. The basis of the method is to simply ‘cut’ the cell where it intersects the

boundary surface and discretize the governing equations on the truncated cell (see Fig. 1.6).

While simple in concept, in practice discretizing the governing equations on complex poly-

hedral cells is not trivial. Additionally, poorly formed cells as a result of the truncation, in

particular small cells with high aspect ratios (Fig. 1.6), can impart excessive stability con-

straints on the incremental solution step size, greatly reducing efficiency. In order to address

these issues, a cell-merging [66, 67, 76, 77] or cell-linking [78] strategy is typically employed.

More recently, virtual-merging has been used [73, 79], which replaces the merging/linking proce-

dure by instead transferring mass/momentum sources of cells that are too small to neighbouring

non-small cells. This method greatly simplifies the data structures necessary for implementa-

13

tion, can ensure mass conservation, and has been shown to suppress SFOs by approximately

an order of magnitude.

Indirect Forcing

Indirect forcing methods aim to derive a momentum body forcing term to impose a specific

boundary condition. In essence, indirect forcing is similar to the CF methods discussed in

Section 1.4.1, but rather than compute forcing from the deformation of an elastic body, an

a priori estimate of the forcing is made using information from the numerical solution, and

thus forcing is applied after discretization. The first documented use of indirect forcing was

the virtual boundary method of Goldstein et al. [80], which employed a forcing term governed

by a feedback loop. Transfer operators similar to Eq. (1.5) are used to spread forces from

Lagrangian to Eulerian points and vice versa, and can also be used to transfer fluid velocities.

The discrepancy between the interpolated velocity and target boundary condition can then

be used to compute a feedback force to impose the desired boundary condition. The velocity

interpolation onto the Lagrangian points would take the form,

U(X, t) =∑i

Di(x)ui(xi, t) (1.9)

where U is the interpolated velocity at the Lagrangian point, and Di is the spreading operator.

A feedback force is then computed such that U satisfies the boundary condition, and the force

is distributed back to the Eulerian grid points via Eq. (1.5). Many choices for the operator D

exist. For example, in one of the earlier iterations of this method, Saiki and Biringen [81] used

a bilinear interpolation operator, while Ren et al. [82] use the same continuous kernel used by

Lai and Peskin [62] to perform interpolation and force spreading. When solving the Navier-

Stokes equations, they first perform a predictor step in order to obtain a predicted velocity

without forcing. A velocity correction is then computed such that the boundary conditions at

the Lagrangian points are satisfied, and the force required to enforce the boundary condition is

distributed back to the Eulerian grid points using the spreading operator.

More recently, Fadlun et al. [83] introduced embedded boundary methods, which share

characterizations of both direct and indirect methods. A forcing term is still utilized to enforce

14

the boundary condition, but rather than use spreading operators described previously to spread

the force in a continuous manner over the width of a kernel, a local reconstruction is instead

performed on cells near the fluid-solid interface that requires only that the boundary condition

be respected. The cell classification for a typical embedded boundary method is displayed in

Fig. 1.7, though cell classification may vary depending on the interpolation type chosen.

Fluid cell Solid cell IB cell

Ωib

Γib

Figure 1.7: Schematic showing a generic body defining an IB domain, occupying volume Ωib

with boundary Γib. The cells are flagged using a common flagging scheme in many DF IBMs.

In [83], the reconstruction of the velocity near the boundary is performed simply using a

one-dimensional linear interpolation along grid lines. Balaras [84] used bilinear interpolation to

determine the velocity at a virtual point on a line oriented normal to the boundary surface, and

then performed linear interpolation along the line. Later, Yang and Balaras [85] improved this

approach by implementing a multi-dimensional linear interpolation, and demonstrated second

order accuracy for flow around fixed and moving cylinders. Embedded methods retain a sharp

interface by localizing the forcing to cells adjacent to the IB, and therefore can also be classified

as sharp-interface methods. The approach of formulating the forcing term explicitly, as opposed

to the direct methods described in Section 1.4.2, has several advantages when dealing with

moving boundary problems. Issues regarding geometric conservation [73] are avoided since the

15

governing equations are solved on the entire domain, and boundary conditions for the pressure

Poisson equation are not needed. Additionally, solving the governing equations on the entire

domain ensures a greater degree of continuity of the velocity field.

1.5 Thesis Objective

The objective of this thesis is to develop a coupled VOF and IBM numerical framework capable

of simulating fixed and moving IBs, with the ability to prescribe a wide range of contact angles.

Little work has been done in coupling IBMs and multiphase flow solvers, thus this thesis aims to

develop capabilities that do not currently exist through new algorithms, as well as coupling pre-

existing IBM and VOF methodologies. A fully parallelized, object-oriented code was developed

for the incompressible, multiphase Navier-Stokes equations with variable density and viscosity,

and new IB formulations are proposed to specify contact angle boundary conditions for the

VOF field. The numerical models are applied to a wide range of problems ranging from droplet

impacts onto cylinders, moving particles at interfaces, to the injection of a fluid into a porous

media. The main sub-objectives of this work include:

• Couple a VOF interface advection algorithm with a fixed IBM. Special attention is given

to ensure mass conservation errors, where applicable, are minimized.

• Implement a method of imposing a wide range of contact angles on static IBs. Validate

the proposed methods on IBs of various shapes, by ensuring that the correct equilib-

rium contact angles are recovered for droplets initially perturbed from the equilibrium

configuration.

• Extend the approach to moving IB problems. Add the effect of the capillary forces on

the body.

• Validate the numerical models against select theoretical and experimental results obtained

from the literature, in order to demonstrate the correctness of the models.

• Apply the models to select problems, including the interaction of particles undergoing

solid-body motion at an interface, and flow through a porous media, in order to demon-

16

strate potential applications of the proposed models.

1.6 Thesis Organization

The remainder of this thesis is structured as follows. In Chapter 2, the governing equations

and mathematical formulations on which the numerical models are based are given. In Chapter

3, details pertaining to the discretization of the governing equations and their implementation

into code is given. In Chapter 4, a contact line model for ghost-cell IBs is presented, which

is validated by solving various problems involving droplets interacting with stationary IBs. In

Chapter 5, an indirect forcing IBM is presented, which is coupled to a similar contact line model.

A range of FSI problems which include capillary forces are solved in order to demonstrate the

efficacy of the algorithm. In Chapter 6, the method developed in Chapter 5 is applied to a

porous media flow. Finally, in Chapter 7, the thesis is summarized, with particular attention

given to the contributions of this work, and some final thoughts on potential directions for

future research.

17

Chapter 2

Governing Equations

In this chapter, the governing equations and mathematical formulations on which the numerical

models are based are presented. The incompressible Navier-Stokes equations are used to model

immiscible two-phase flow with variable density and viscosity, and are discussed in section 2.1.

The non-dimensionalized equations are presented in Section 2.2.

2.1 Conservation Equations

The incompressible Navier-Stokes equations for two-phase flows consist of the conservation of

mass and momentum equations. In vector notation, conservation of mass is written as,

∂ρ

∂t+∇ · ρu = 0 (2.1)

where ρ is the fluid density and u is the local flow velocity. In this work, ρ = ρ(γ), where γ is

the volume fraction. The conservation of momentum equation takes the form,

∂ρu

∂t+∇ · ρuu =∇ · τ −∇p+ ρg + fσ (2.2)

where g is the gravitational acceleration, fσ is the surface tension body force and τ is the viscous

stress tensor, which for a Newtonian fluid takes the form,

τ = µ(∇u +∇uT

)(2.3)

18

Eqs. (2.1) and (2.2) are not a closed system of equations, since for incompressible flows

there is no equation of state to relate the primitive variables ρ and p. Thus, in order to close

the system of equations, the divergence-free constraint is imposed on the velocity field,

∇ · u = 0 (2.4)

The discussion of how to use the p field to enforce Eq. (2.4) is reserved until section 3.2.

Fluid properties such as density and viscosity are evaluated as functions of the volume fraction,

γ,

γ =

0 inside fluid #1

0 < γ < 1 interface

1 inside fluid #2

(2.5)

Under the assumption of incompressible flow, no equation of state exists that relates the

fluid properties to a thermodynamic state, and therefore fluid properties such as ρ and µ are

functions of γ only. For local average density, simple linear weighting of the form,

ρ(γ) = ρ1 + (ρ2 − ρ1)γ (2.6)

can be used, where the subscripts 1 and 2 signify the properties of the first and second fluids

respectively (a harmonic average has been shown to reduce errors in simple multiphase flows

when high density ratios are present [86]). While linear weighting can also be used to compute

dynamic viscosity µ, stability issues can arise at higher viscosity ratios. Thus, Prosperetti

[87] suggested a more fundamental approach by taking the harmonic average of the kinematic

viscosity,

ρ(γ)

µ(γ)=ρ1

µ1+

(ρ2

µ2− ρ1

µ1

)γ (2.7)

Substitution of Eq. (2.6) into Eq. (2.1) yields the VOF advection equation,

∂γ

∂t+∇ · uγ = 0 (2.8)

19

Thus, Eq. (2.8) serves as both the mass conservation equation, as well as the colour function

advection equation Eq. (1.1). The surface tension force, fσ in Eq. (2.2), is also a function of

γ, and is derived using the CSF method of Brackbill et al. [51]. The surface tension force is

defined by,

∫A

fσa(xs) dA = limh→0

∫Vσκ(x)

∇C∆C

δ[n · (x− xs)] dV (2.9)

where fσa is the surface tension force per unit interface area, h is the interface thickness,

∆C = C2 − C1, and xs is a position on the interface. Note that in Eq. (2.9), the choice of the

colour function C can be chosen as γ, in which case the per unit volume force simplifies to,

fσ = σκ∇γ (2.10)

Alternatively, one can choose C = ρ in Eq. (2.9). If one multiplies Eq. (2.9) by g(x) =

C(x)/C where C is the arithmetic mean of C, the value of the limit is not changed since at

x = xs, g(xs) = 1. Thus, Eq. (2.10) can be alternatively written as,

fσ = σκ∇ρ∆ρ

ρ

ρ(2.11)

where ρ is the mean density. This form has some advantages in that it decouples fσ from the

absolute value of density, thus heavier fluid elements in a transient region will experience the

same acceleration as lighter fluid elements. As noted in [51], this prevents an interface from

thickening/thinning depending on whether fσ points to the lighter or denser fluid, which can

be a source of spurious oscillations.

In order to simplify the boundary conditions for p when buoyancy forces are taken into

account, p is replaced by a piezometric pressure p similar to the approach of Montazeri et

al. [88]. This pseudo-pressure is defined as,

p = p− ρg · x (2.12)

where x in this context refers to an arbitrary position in the Eulerian reference frame. Taking

the gradient of Eq. (2.12) and re-arranging the terms leads to the relationship,

20

−∇p+ ρg = −∇p−∇ρg · x (2.13)

which can then be substituted into Eq. (2.2) to yield,

∂ρu

∂t+∇ · ρuu =∇ · τ −∇p−∇ρg · x + fσ (2.14)

The effect of this substitution is to transform the buoyancy force into an interface-only

force, which is zero everywhere except where a density gradient is present. This term can be

combined with Eq. (2.11) into a single interface force dependent only on the gradient of density

and interface curvature. Grouping the interface terms in such a manner ensures a consistent

discretization of the body forces, which is critical for minimizing spurious velocities [48, 52].

2.2 Non-Dimensionalization

The non-dimensional form of Eq. (2.2) is,

∂λρu∗

∂t+∇ · λρu∗u∗ =

1

Re∇ · τ∗ −∇p∗ +

λρ

Fr2 g +1

Wef∗σ (2.15)

where λρ is the density ratio, g is the gravitational unit vector, and the superscript ∗ denotes

a non-dimensionalized quantity. The Reynolds number Re is defined as,

Re =ρ|U|Lµ

(2.16)

where L is a characteristic length. The Froude number Fr is defined as,

Fr =|U|√|g|L

(2.17)

The Weber number We is defined as,

We =ρ|U|2Lσ

(2.18)

the Eotvos number Eo or Bond number Bo, which is useful when buoyancy is involved, is

21

defined as,

Eo = Bo =∆ρgL2

σ(2.19)

where ∆ρ is the difference in density between two fluids. The capillary number Ca, which

represents the ratio of viscous and capillary forces is defined as,

Ca =µ|U|σ

(2.20)

Finally, the Ohnesorge number Oh, which represents the ratio of viscous to inertial and

capillary forces is expressed by,

Oh =µ√ρσL

=Ca√We

=

√We

Re(2.21)

Additionally, some cases are better parameterized by non-dimensional numbers other than

those in Eq. (2.15). By combining Eqs. (2.16), (2.17), (2.18), (2.20) and (2.21), additional forms

of Eq. (2.15) can be obtained with parameters that better characterize the problem of interest.

For example, in the context of porous media flows, it is convenient to cast the momentum

equation in the form,

Re

(∂λρu

∗

∂t+∇ · λρu∗u∗

)=∇ · τ∗ −∇p∗ +

Reλρ

Fr2 g +1

Caf∗σ (2.22)

since such flows are better characterized by Ca. Note that the definitions of the non-dimensional

viscous stress tensor τ∗ and the non-dimensional pressure p∗ have changed in Eq. (2.22).

22

Chapter 3

Numerical Methodology

In this chapter, the discretizations of the governing equations presented in Chapter 2 is pre-

sented. In Section 3.1, the momentum equation used in the fractional-step procedure is dis-

cretized; in Section 3.2, closure of the discretized Navier-Stokes equations is presented using the

projection method of Chorin [89]; in Section 3.3, the compressive interface advection scheme

used to discretize Eq. (2.8) is discussed. The strategies for computing the surface tension force

are presented in section 3.4. The implementation of IBMs is presented in section 3.5. Remarks

regarding the coupling of IBMs and VOF are made in Section 3.6, however, most of the details

of these models are reserved for chapters 4 and 5. Finally, strategies for solving the large, sparse

systems of equations arising from the discretization of the governing equations in parallel are

covered in section 3.7.

3.1 Discretization of the Momentum Equation

3.1.1 Fractional-Step Method

The momentum equation, Eq. (2.14), is solved using the fractional-step projection method for

a collocated variable arrangement. In this work, a two-step predictor-corrector scheme is used

to solve for the velocities at the cell-centers, uc, which proceeds as follows,

23

1. Predictor step:

ρn+1c u∗c − ρncunc

∆t+

1

2

[Fn + Fn−1

]=

1

2[D∗ +Dn]− ρn+1

c

⟨∇pρ

⟩nf→c

+ ρn+1c

⟨fg

ρ+

fσρ

⟩nf→c

(3.1)

2. Predictor step:

u∗∗c = u∗c + ∆t

(⟨∇pρ

⟩nf→c−⟨

fg

ρ+

fσρ

⟩nf→c

+

⟨fg

ρ+

fσρ

⟩n+1

f→c

)(3.2)

3. Corrector step:

un+1c = u∗∗c −∆t

⟨∇pρ

⟩n+1

f→c(3.3)

The superscripts n and n + 1 denote the time-levels at which each term is evaluated, the

superscript ∗ denotes an intermediate predicted value, F , D are the convective and diffusive

fluxes respectively, and the operator 〈·〉f→c denotes an appropriate face-to-cell interpolation of

a body forcing term [48]. The body forces fg and fσ, as well as properties ρ and µ, will already

be available at the n+ 1 time level at the time Eq. (3.1) is solved since they are functions of γ,

which is solved before momentum. Following the balanced force procedure of Francois et al. [48],

since pressure is only available at the n time level and these body forces are discontinuous at

the interface, the previous values of fg and fσ are used in Eq. (3.1) so that they are balanced

appropriately by ∇pn. ∇pn+1 must be obtained from the projection step, which is discussed in

section 3.2.2. In order to compute the divergence of the velocity field, which is needed in order

to compute the pressure Poisson equation (section 3.2.2), velocity is interpolated to cell faces.

A procedure similar to the well-known Rhie-Chow momentum interpolation [90] is used, which

gives,

24

u∗∗f =α

(u∗cl +

⟨∇pρ

⟩nf→cl

−⟨

fg

ρ+

fσρ

⟩nf→cl

)

+ (1− α)

(u∗cr +

⟨∇pρ

⟩nf→cr

−⟨

fg

ρ+

fσρ

⟩nf→cr

)

+∆t

ρf

(fn+1σ + fn+1

g

)f

(3.4)

where α denotes the linear interpolation weighting, usually either by inverse distance or volume,

and cl and cr denote the cells to the left and right of the face respectively.

The fluxes from Eq. (3.1) must also be computed. A Crank-Nicolson procedure is used. The

convective flux is evaluated as,

F =1

Ωc

∑f

(ρuu)f · sf (3.5)

where the subscript f denotes a quantity evaluated at the centre of the cell face, Ωc is the

volume of the cell under consideration, and sf is an outward pointing vector which is normal

to the cell face, and has a magnitude equal to the face area. Similarly, the diffusive flux is

evaluated using the method of Mathur and Murthy [91] for orthogonal meshes,

D = − 1

Ωc

∑f

µf (unb − uc)rnb · sfrnb · rnb

(3.6)

where rnb is a relative vector which points to the neighbouring cell centre from the cell centre

under consideration. Note that in Eq. (3.1), a portion of the diffusive flux is handled implicitly.

While this does introduce some error into the projection step, computing D semi-implicitly helps

avoid excessive time step constraints, since for a purely explicit scheme the time step would be

bounded by ∆t < ∆h2

2µ , where ∆h is the distance between cell centers for an equidistant mesh.

3.1.2 Evaluation of Body Forces

In this section, the discretization of the body forces in Eq. (3.1) is presented. In general, in

order to avoid spurious force generation as a consequence of an inconsistent discretization of

body forces [48, 88], body forces must be discretized in a consistent fashion, and the represen-

25

~sf

~rc

Figure 3.1: Schematic of a typical FVM cell on which discretized Navier-Stokes equations aresolved.

tation of body forces at cell faces must be consistent with their representation at cell centres.

Additionally, body forces which are averaged over an entire cell volume should be computed in

a way that is consistent with their representation on cell faces. Francois et al. [48] proposed

the following face-weighted averaging scheme for the components of a vector field,

xi · 〈f〉f→c =

∑f (xi · ff )|xi · sf |∑

f |xi · sf |(3.7)

where xi denotes the ith cartesian basis vector. Finally, the normal components of the face-

centred gradients on orthogonal meshes can be computed from,

∇φf ≈ (φnb − φc)rnb

rnb · rnb(3.8)

3.2 Chorin’s Projection Method

The equations Eq. (2.1) and Eq. (2.2) do not represent a closed system of equations. When

dealing with incompressible fluids, no equation of state is available to close the system by

relating the primitive variables ρ and p. Thus, in order to close the system and obtain an

additional equation, the divergence-free constraint (Eq. (2.4)) is imposed. Eq. (2.4) does not

26

contain p, however, so in order to close the system, a strategy to relate u and p known as the

projection method is employed, and is discussed in this section.

3.2.1 Helmholtz–Hodge Decomposition

The basis of the projection method is the so-called Helmholtz-Hodge decomposition, which

states that a sufficiently smooth vector field can be broken down into irrotational and divergence-

free components. Thus, in Eq. (3.2), the second predicted velocity field u∗∗, which is neither

irrotational nor divergence-free, can be decomposed into,

u∗∗ = u + uirrot = u +∇φ (3.9)

where u and uirrot are the divergence-free and curl-free components of u∗∗ respectively, and φ

is a scalar field, the gradient of which can be substituted for uirrot. Taking the divergence of

Eq. (3.9), it follows that,

∇ · u∗∗ = ∇2φ (3.10)

which is the well-known Poisson equation for φ, which can be solved with an appropriate choice

of boundary conditions.

3.2.2 Pressure Poisson Equation

Evidently, if u∗∗ is known from solving Eq. (3.1) and (3.2), and φ is obtained from solving Eq.

(3.10), the divergence-free velocity un+1 can be obtained from,

un+1f = u∗∗f −∇φf (3.11)

un+1c = u∗∗c − 〈∇φ〉

n+1f→c (3.12)

Substituting Eq. (3.12) into Eq. (3.2), and then Eq. (3.2) into Eq. (3.1) yields,

27

ρn+1c un+1

c − ρncunc∆t

+1

2

[Fn + Fn−1

]=

1

2[D∗ +Dn] + ρn+1

c

⟨fg

ρ+

fσρ

⟩nf→c

+ρn+1

∆t〈∇φ〉n+1

f→c

(3.13)

and it can then be deduced from inspection of Eq. (3.13) that φ can be related to the pressure

field by the equation,

〈∇φ〉f→c = ∆t

⟨∇pρ

⟩f→c

(3.14)

Thus, the pressure Poisson equation becomes,

∇ ·[

∆t

ρn+1∇p]

=∇ · u∗∗ (3.15)

which, in discrete form, becomes,

∑f

∆t

ρn+1f

(pnb − pc)rnb · sfrnb · rnb

=∑f

u∗∗f · sf (3.16)

Upon solving Eq. (3.15), the predicted velocities u∗∗ are corrected at both cell and face

centers,

un+1c = u∗∗c −∆t

(⟨∇pρ

⟩n+1

f→c−⟨

fg

ρ+

fσρ

⟩n+1

f→c

)(3.17)

and,

un+1f = u∗∗f −

∆t

ρn+1f

∇pn+1f (3.18)

The choice of boundary conditions for Eq. (3.15) is relatively straightforward. If the bound-

ary represents a surface through which there is a fixed flux, the second term on the right-hand-

side of Eq. (3.18) must be zero, thus,

∇pf · nw = 0 on ∂Ωb (3.19)

28

For other types of boundary conditions, such as outflow boundary conditions, it is acceptable

to prescribe,

pf = c on ∂Ωb (3.20)

and apply the normal correction procedure, Eq. (3.18).

3.3 A Consistent CICSAM Scheme for Computing Mass and

Momentum Flux

In this work, the CICSAM scheme of Ubbink and Issa [41] is used to compute the fluxes in Eq.

(2.8), which provides a good balance between efficiency and maintaining a shaprp interface.

Additionally, careful consideration is given to ensure the resulting mass fluxes obtained from

Eq. (2.8) are exactly consistent with the momentum flux used to advect u∗ in Eq. (3.1), which

as noted by Bussmann et al. [92] is critical for flows with high density ratios. In this section we

describe the method used to ensure consistency between the discretizations of Eq. (3.1) and Eq.

(2.8). First, the value of γ for a donor cell is normalized according to the Normalized Variable

Diagram (NVD) of Leonard [93],

γD =γD − γUγA − γU

(3.21)

where γU , γD and γA are the values of γ at the upwind, donor and acceptor cells respectively.

As proposed by Jasak et al. [94], the value of γU can be computed as,

γU = γA + 2(xD − xA) ·∇γD (3.22)

where xD and xA are the position vectors of the donor and acceptor cells respectively. The two

schemes used to compute the normalized value of γ at the face are the Hyper-C (HC) scheme,

γHCf =

min 1, γD

CoDwhen 0 ≤ γD ≤ 1

γD otherwise

(3.23)

29

and the Ultimate Quickest (UQ) scheme [93],

γUQf =

min 8CoD γD+(1−CoD)(6γD+3)

8 , γHCf when 0 ≤ γD ≤ 1

γD otherwise

(3.24)

The HC scheme has a tendency to artificially distort the interface as it moves tangentially

to cell faces, while the UQ scheme is overly dissipative. In order to remedy this, a blending

function is employed to combine both schemes, such that,

γf = ψf γHCf + (1− ψf )γUQf (3.25)

where ψf is a weighting function that depends on the interface orientation. Ubbink and Issa

[41] proposed the following form for ψf ,

ψf = min

kψ

cos (2θf ) + 1

2, 1

(3.26)

where the angle θf , which defines the angle between the interface and the relative vector con-

necting the donor and acceptor cells, is defined as,

θf = arccos

∣∣∣∣n · ~rA|~rA|∣∣∣∣ (3.27)

The constant kψ adjusts the relative dominance of Eqs. (3.23) and (3.24). Ubbink and Issa

[41] proposed a value of kψ = 1, which is used in this work. Once γf is determined from Eq.

(3.25), it is used to compute a weighted linear interpolation parameter β from the equation,

β =γf − γD1− γD

(3.28)

and the colour function value γf is interpolated as,

γf = γD + β(γA − γD) (3.29)

which is then used as the face centered value in the discretized form of Eq. (2.8). Additionally,

the momentum flux term F in Eq. (3.1) becomes,

30

F = (ρ1 + (γD + β(γA − γD))∆ρ)[max(0,uf · sf )uc + min(0,uf · sf )unb] (3.30)

when an upwind advection scheme is used for momentum. Higher-order upwind schemes can

also be accomodated in a similar fashion, but are omitted here in the interest of brevity.

Note that Eq. (3.30) effectively means a different value of ρf is used for advecting momentum

than is used to solve the Poisson equation. Using this interpolation method alongside the density

weighted face-to-centre evaluation of the pressure gradient described by Francois et al. [48] has

enabled us to simulate flows of density ratios up to 1012 without need of harmonic interpolation

of densities at faces, or computing the densities based on a convolved γ field as suggested in

[95].

3.4 Computation of Surface Tension

In the CSF method of Brackbill et al. [51], the interface force is fσ = σκ∇γ, with the curvature

κ computed as,

κc =∇ · n (3.31)

where n is the interface unit normal vector, defined as,

n = − ∇γ‖∇γ‖ (3.32)

where γ is a convolved volume fraction field, explained in the following section.

3.4.1 Computation of the Interface Normals

The unit normal vector to the interface is computed directly from a smoothed γ field. Using the

actual γ field leads to large errors due to the discontinuous nature of the interface. Numerous

authors [51, 96] have noted that these errors can be greatly reduced by computing the gradient

based on a convolved γ field instead, denoted as γ. Thus, for the purpose of evaluating the

interface normals, a convolved volume fraction field γ is computed as

31

γ =

∫ΩK(x)γdΩ (3.33)

where the kernel function K(x) is a simplified version of the one used by Peskin [59], which

takes the form

K(x′) =

A

2rk(1 + cos(πr/rk)) if r < rk

0 otherwise

(3.34)

where rk is the effective radius of the kernel, A is a constant selected such that the kernel is

normalized to unity, and r = |x′ − x|. Through experimentation, it was found that a kernel

radius of rk = 2∆l, where ∆l is a characteristic cell length, resulted in the smallest parasitic

currents. In order to compute ∇γ in the CELESTE scheme [97], the Taylor series of γ is

expanded to second-order from neighbouring cells to the cell centre of interest in the following

manner:

γc = γn +∂γn∂x

∆x+∂γn∂y

∆y +∂2γn∂x2

∆x2

2+∂2γn∂y2

∆y2

2+∂2γn∂x∂y

∆x∆y +O(∆x3,∆y3) (3.35)

Eq. (3.35) can be arranged into a weighted, over-determined linear system of the form Ax = b,

in terms of the unknown derivatives, yielding

∆x1∆s21

∆y1∆s21

∆x212∆s21

∆y212∆s21

∆x1∆y1∆s21

......

......

...

∆xi∆s2i

∆yi∆s2i

∆x2i2∆s2i

∆y2i2∆s2i

∆xi∆yi∆s2i

......

......

...

∆xN∆s2N

∆yN∆s2N

∆x2N2∆s2N

∆y2N2∆s2N

∆xN∆yN∆s2N

∂γ∂x

∂γ∂y

∂2γ∂x2

∂2γ∂y2

∂2γ∂x∂y

=

∆γcn1

∆s21...

∆γcni

∆s2i...

∆γcnN

∆s2N

(3.36)

Haselbacher and Vasilyev [98] identified two constraints on the use of least-squares methods

for reconstructing a gradient from a Taylor series: (1) the stencil should be as small as possible

to reduce truncation error, and (2) the stencil should be as symmetric as possible in order to

cancel out truncation error terms. Since at least five cells are needed to solve Eq. (3.36), the

32

cell face and diagonal neighbours are used, resulting in the 3× 3 cartesian stencil shown in Fig.

3.2.

Figure 3.2: Stencil used for the least-squares reconstruction of the interface in the CELESTEscheme [97].

As per Fig. 3.2 (b), when adjacent cell neighbours are not available, for example along bound-

aries, the corresponding face or node is used instead, with the value of γ depending on the

boundary condition.

3.4.2 Computation of Interface Curvature

The procedure for computing the curvature follows a similar process to the interface normals.

The curvature can be expressed as the divergence of the interface unit normals,

κ =∇ · n =∂nx∂x

+∂ny∂y

(3.37)

A linear system is then solved for each component of n, denoted by the subscript j, taking the

form,

33

∆x1 ∆y1∆x21

2∆y21

2 ∆x1∆y1

......

......

...

∆xi ∆yi∆x2i

2∆y2i

2 ∆xi∆yi...

......

......

∆xN ∆yN∆x2N

2∆y2N

2 ∆xN∆yN

∂nj

∂x

∂nj

∂y

∂2nj

∂x2

∂2nj

∂y2

∂2nj

∂x∂y

=

∆nj,cn1

...

∆nj,cni

...

∆nj,cnN

(3.38)

where nj is a component of the unit normal vector n. Our numerical experiments have shown

consistently that more accurate curvatures are achieved when using an unweighted least-squares

procedure for the computation of interface normal divergence. Upon solving Eq. (3.38), the

appropriate derivatives are substituted into Eq. (3.37) to obtain the cell centered curvature.

This approach differs slightly from the original approach of Denner and van Wachem [97], in

that for the evaluation of κ, they considered only the first derivatives of n. Overall, the use of

higher order derivatives was found to produce a smoother curvature field and fewer parasitic

currents for the cases studied here.

Finally, unlike the original CELESTE method, no smoothing of the curvature was per-

formed. Instead, the curvature was simply not evaluated in cells where ‖∇γc‖ < ε, where ε is

a small cut-off value. For interpolating curvatures to faces, ‖∇γf‖ < ε was also enforced, even

if the cell centred gradient was non-zero.

3.5 Implementation of the Immersed Boundaries

In this section, the implementation of the immersed boundary method will be shown. Both a

direct and an indirect approach are discussed. The direct approach using a ghost-cell method

is described in Section 3.5.1, while an indirect approach more suitable for moving boundary

problems is discussed in Section 3.5.2.

3.5.1 Ghost-Cell Direct Approach

As previously mentioned in Section 1.4.2, the ghost-cell IBM is a direct-forcing method that

imposes boundary conditions on an arbitrary scalar field φ such that when interpolated onto

∂Ωb, φ = ∂φb is recovered. The ghost-cell method was made popular by Tseng and Ferziger

34

[68], where a symmetric interpolation stencil is constructed across the boundary as shown in

Fig. 3.3.

IB cell Fluid cell

Stencil A Stencil B

xbp

xip

xbp

xip

Figure 3.3: Interpolation stencil used in the ghost-cell immersed boundary method, showingthe image point xip and boundary point xbp.

In the method used in this work, two candidate stencils A and B are constructed. For

the case of stencil A, the image point xip is close to the boundary, and thus the value of φip

is a function of φgc. For stencil B, φgc is not needed to compute φip. As shown in the next

subsection, the two stencils are needed in order to deal with the possibility of singular values

when performing a polynomial interpolation of φip. In general, φip is computed as,

φip = a3xy + a2x+ a1y + a0 (3.39)

where the coefficients a0–a3 are unknown, and must be computed from the values of φ within

the cells that bound xip. This can be done by forming the Vandermonde matrix [99],

1 y0 x0 x0y0

1 y1 x1 x1y1

1 y2 x2 x2y2

1 y3 x3 x3y3

a0

a1

a2

a3

=

φ0

φ1

φ2

φ3

(3.40)

or,

35

Aa = φ (3.41)

where xi and yi are the cartesian coordinates of the ith cell centre. We can manipulate Eq.

(3.41) in order to obtain an expression of φip in terms of the linear combination φip =∑αiφi

by multiplying through the inverse of A and the row vector x,

φip = xa = xA−1φ (3.42)

where,

x =

(1 yip xip xipyip

)(3.43)

where xip and yip are the cartesian coordinates of the image point x. One advantage of the direct

forcing method is that φip in Eq. (3.42) can be incorporated into the discretized momentum

equation implicitly, eliminating the need to perform iterative corrections. Additionally, the

extension of the calculation of φip from 2D to 3D is straightforward, since it requires only

replacing the bilinear interpolation in Eq. (3.40) with a trilinear interpolation.

Imposing Boundary Conditions

In the previous section, it was shown how the value φip could be recovered by choosing an

appropriate interpolation scheme. In general, three types of boundary conditions exist for

φ: fixed value (Dirichlet), fixed normal gradient (Neumann), or a linear combination of both

(Robin). The approach to imposing the boundary condition depends on whether the stencil is

of type A or B. For a Dirichlet boundary condition,

Stencil A:

φgc = xgc

1 y0 x0 x0y0

1 y1 x1 x1y1

1 y2 x2 x2y2

1 ybp xbp xbpybp

−1

φ0

φ1

φ2

φbp

(3.44)

36

Stencil B:

φgc = 2φb − xipA−1φ (3.45)

Similarly, for a Neumann boundary condition of the form,

∂φ

∂n= c (3.46)

where n is the normal coordinate and c is a constant, the boundary condition take the form,

Stencil A:

φgc = xgc

1 y0 x0 x0y0

1 y1 x1 x1y1

1 y2 x2 x2y2

0 ny nx ygcnx + xgcny

−1

φ0

φ1

φ2

c

(3.47)

Stencil B:

φgc = ∆lc+ xipA−1φ (3.48)

where ∆l is the length of the stencil. The purpose of using two different stencils depending

on whether or not the bounding cells of the image point contain the image point is to prevent

singular systems of equations. For example, if stencil B is used when computing a fixed gradient

boundary condition, using Eq. (3.48) results in xgc → xip, which cannot be used to compute

the normal gradient using a finite difference. Additionally, for fixed boundary conditions using

stencil A, Eq. (3.44) will become singular if xbp lies too close to one of the bounding cell centres

in the fluid.

The above procedure is widely known to be second-order for Dirichlet boundary conditions,

consistent with the order of the bilinear interpolation function, but reduces to first-order for

enforcing Neumann boundary conditions [70]. This leads to locally first-order approximations

of the pressure gradient. The pressure however, being an integral of the pressure gradient,

retains second order accuracy. Additionally, although the divergence-free velocity is a function

of the pressure gradient as shown by Eq. (3.17) and (3.18), the gradient gets multiplied by ∆t,

and since O(∆t) ∝ O(h) where h is the mesh spacing, the corrected velocity field retains local

37

second order accuracy near the boundary.

3.5.2 Piecewise Quadratic Indirect Approach

The piecewise quadratic indirect approach is an alternative IBM strategy. It falls under the

category of indirect forcing methods previously introduced in section 1.4.2. In this approach,

boundary conditions are not prescribed as primitive variables inside the boundary, but rather a

forcing term, fib, is included in Eq. (2.2) as a body forcing term. This approach has numerous

advantages. For one, boundary conditions do not necessarily need to be prescribed for pressure,

but rather the pressure Poisson equation can be solved over the entire domain. This alleviates

the issue of geometric conservation that can lead to SFOs during moving boundary simulations

[73, 74], since the governing equations are solved on the entire domain rather than just the fluid

domain. Secondly, there is no need for different stencils as in the ghost-cell method; rather,

the stencil is always well-defined. The method of constructing the forcing term in this work

is similar to the approach of Mohd-Yusof [100]. In this procedure, an additional momentum

predictor step is performed in between Eqs. (3.1) and (3.2). Initially, a predictor step identical

to Eq. (3.1) is performed, which does not include the effect of the IB. A forcing term is then

calculated, which takes the form,

ρn+1(u∗ − uib)

∆t= fib (3.49)

where uib is a velocity that satisfies the prescribed boundary condition. Note that for regions

away from the influence of the IB, u∗ = uib. And inside the IB itself, uib can simply be

prescribed to the solid body motion of the IB. In the forcing region, defined by the IB cell

classification depicted in Fig. 1.7, a reconstruction must be used such that the nearby boundary

condition is respected. In this scheme, it is prudent to assume that the local flow near the

boundary varies in a biquadratic fashion. In order to perform the reconstruction, nearby fluid

cells, IB cells and the boundary condition itself are used to construct an interpolation stencil,

as shown in Fig. 3.4.

By expanding a biquadratic polynomial from the centre of the forcing cell to the reconstruc-

tion points, a system of equations for fib can be constructed which takes the form,

38

(xj , yj)

uib

(xi, yi)

Figure 3.4: Least-squares stencil (red) constructed by expanding a biquadratic polynomial toneighbouring fluid and forcing points. Dark gray cells are solid cells, light gray cells are forcingcells. The dashed blue lines denote neighbouring stencils.

∆t

ρn+1fib =

∑βi

(∆t

ρn+1fib + u∗

)i

+∑

βjuib + u∗ (3.50)

where β is an interpolation coefficient and the subscripts i and j denote the ith and jth cell

and boundary interpolation points respectively. Similar to the ghost-cell approach described

in the previous section, the interpolation coefficients β can be computed by constructing a

Vandermonde matrix of the form,

β =

x2c

y2c

xcyc

xc

yc

1

T

x2i,1 y2

i,1 xi,1yi,1 xi,1 yi,1 1

......

......

......

x2i,Ni

y2i,Ni

xi,Niyi,Ni xi,Ni yi,Ni 1

x2j,1 y2

j,1 xj,1yj,1 xj,1 yj,1 1

......

......

......

x2j,Nj

y2j,Nj

xj,Njyj,Nj xj,Nj yj,Nj 1

−1

(3.51)

where (xc, yc) are the cartesian coordinates of the forcing cell, (xi, yi) are the cell centre coor-

39

dinates of the neighbouring fluid and forcing cells, and (xj , yj) are the boundary points.

The general solution procedure proceeds as follows:

1. Solve the first momentum prediction, Eq. (3.1)

2. Compute the forcing term from Eq. (3.50)

3. Solve the second momentum predictor, Eq. (3.2)

4. Solve the pressure Poisson equation, Eq. (3.15)

5. Perform the projection step, Eqs. (3.17) and (3.18)

Generally, in indirect approaches, u∗ is not smooth across the boundary. An extensive

discussion regarding the spatial accuracy of IBMs in which the projection is performed over the

entire domain is given by Guy and Hartenstine [101]. They showed that due to the discontinuous

nature of u∗ across the boundary, the order of accuracy was O(h) with respect to max-norms,

but remained O(h2) with respect to integral norms.

3.6 Coupling of IBMs and VOF

The coupling of IBMs and VOF involves extending the γ into the boundary along the contact

line (similar to the fixed mesh approach of Afkhami [102]), and computing modified curvatures

near the surface such that the known contact line normal is respected. With both of the IBMs

described in sections 3.5.1 and 3.5.2, it is also possible to compute mixed or Robin boundary

conditions which allow for partial slip conditions, which are sometimes used in contact line

simulations to help alleviate the stress singularity [55]. Here, the details of the contact line

models, which vary slightly for each IBM, is deferred until chapters 4 and 5.

3.7 Strategies for Solving the Discretized Equations

In this section, the strategies for solving Eq. (3.1) and (3.15) will be briefly described. In

Section 3.7.1, a brief introduction to iterative methods for solving sparse linear systems is

given; in Section 3.7.2, a description of the domain decomposition used for parallel simulations

is given.

40

3.7.1 Solution of the Linear Systems of Equations

The result of the discretization of the governing equations is generally a very large, non-

symmetric sparse system of equations. While these equations could be solved directly for

problems of trivial size, larger systems typically employ an iterative strategy. Some options in-

clude the class of Krylov subspace solvers which include the Bi-Conjugate Gradient STABilized

(BiCGSTAB) [103] method, the Generalized Minimum RESidual (GMRES) [104] method, and

the Algebraic Multi-Grid (AMG) method [105]. A comprehensive overview of various iterative

methods for solving large sparse systems of equations is provided in the book written by Saad

[106]. In this section, the fundamental ideas behind these approaches to solving linear systems

are overviewed, and then for more detailed descriptions the reader is referred to the appropriate

references.

BiCGSTAB Algorithm

The Bi-Conjugate Gradient Stabilized (BiCGSTAB) method is a type of Krylov subspace

method, which can be viewed as a combination of a Bi-Conjugate Gradient immediately fol-

lowed by a GMRES step which is restarted at every step. The motivation behind developing

the BiCGSTAB algorithm was to improve the irregular convergence behaviour of the Conjugate

Gradient Stabilized (CGS) method [103]. The general procedure for the BiCGSTAB iterative

solver, with the convergence tolerance denoted by ε, is given in Algorithm 3.1.

For definitions of the BiCGSTAB constants and vectors in Algorithm 3.1, the reader is

referred to [103]. In general, preconditioners are used in conjunction with BiCGSTAB to

further accelerate convergence. If K is a suitable preconditioning matrix, that is K u A,

then K = K1K2 and,

(K−11 AK−1

2 )(K2x) = (K−11 b) (3.52)

where K1 = I denotes right preconditioning, K2 = I denotes left preconditioning, and K1 = L,

K2 = U denotes preconditioning from both sides. In general, a good choice of K is one which

closely approximates A, but is significantly easier to invert, that is K−1 u A−1. Knoll and

Keyes [107] discuss various approaches to preconditioning within the context of Krylov solvers,

41

Algorithm 3.1: BiCGSTAB algorithm for solving Ax = b.

function bicgstab(A, b, x0, ε)r0 = b−Ax0

ρ0 = α = ω0 = 1for i = 1, 2, . . . , Nmax

ρi = (r, r0)β = ρiα

ρi−1ωi−1

pi = ri−1 + β(pi−1 − ωi−1vi−1)vi = Apiα = ρi/(r0,vi)h = ri−1 + αpis = ri−1 − αvit = Asωi = (t, s)/(t, t)xi = h + ωisif |b−Axi| ≤ ε then

return xielse

ri = s− ωitend

end function

and conclude that effective preconditioning can greatly improve the performance of iterative

Krylov solvers. In the following sections, some common strategies for preconditioning will be

discussed.

Additive Schwarz Preconditioning

In this section, the class of Schwarz preconditioners is discussed. Schwarz preconditioning

is particularly useful in the context of this work, since the preconditioner contains built-in

parallelism and is suitable for solving large systems of equations on distributed parallel High-

Performance Computing (HPC) systems. The basic idea behind Schwarz preconditioning is to

first decompose the computation domain Ω into M smaller parts, known as subdomains, which

are denoted by Ωi, i = 1, 2, . . . ,M , where M also typically corresponds to the number of par-

allel processes used in the simulation. The discretized equations are then solved locally on each

subdomain, and the coupling at the interface of the subdomains in removed in favour of a pre-

scribed boundary condition, either Dirichlet or Neumann. In general, the global preconditioner

K−1 is defined as,

42

K−1 =M∑i=1

PiK−1i Ri (3.53)

where Pi and Ri are the prolongation and restriction operators respectively, which restrict the

global vector to subdomain Ωi, and vice versa. In the case of minimal overlap, the operators

Pi and Ri are not needed, since the required components of the residual vector are already

local. In the case of overlap (Fig. 3.5), each subdomain is identified by a set of local rows

that correspond to the overlapped cells, and the operators Pi and Ri may involve inter-process

communication when assembling Eq. (3.53).

1 2

3 4

1

Dirichlet BCs

Figure 3.5: Graphical depiction of the Schwarz preconditioner with an overlap of one. Theshaded region represents the local block.

Since K−1i is local, its computation proceeds without consideration of the preconditioners on

other processors or blocks. This approach is referred to as additive preconditioning. In general,

in this work an overlap of one was found to produce the best balance between reducing the

number of Krylov iterations necessary to solve the discretized equations, and computational

time required to apply preconditioning. Finally, the use of Schwarz a preconditioner is not

without cost. Due to the decoupling of the preconditioners across processes, the number of

required Krylov iterations generally goes up as M increases. This increase in iterations is

generally offset, however, by the ability to compute the global preconditioner in a parallel

fashion, which was investigated thoroughly by Northrup [108].

43

ILU Preconditioning

In the previous section, a strategy for implementing a global preconditioner for a decomposed

domain was described. In this section, the construction of a common local preconditioner

used in conjuction with Krylov methods is overviewed. The Incomplete Lower-Upper (ILU)

factorization method aims to compute a preconditioner K of the form,

K = LU u A (3.54)

where L and U are the lower and upper factorizations of K respectively. In practice, computing

an exact factorization of A is prohibitively expensive both in terms of time and space, since

even if A is sparse, its LU factorization may not be; which is a phenomenon known as fill-in.

In the ILU approach, approximations of L and U are computed instead; which have pre-

determined sparsity patterns. In the ILU(0) approach, the sparsity patterns are chosen to be

the same as A. Additionally, the accuracy of the preconditioner can be improved by allowing

additional fill. One common choice for the sparsity pattern is to pick the sparsity pattern of

Af+1, which produces the ILU(f) preconditioner, where f is a user specified fill level. In this

work, for diagonally dominant systems of equations arising from the discretizations of unsteady

equations such as Eq. (3.1), f = 0 or f = 1 is generally chosen, since these preconditioners

can be computed very quickly. For f > 1, the time it takes to compute the preconditioner

supercedes any benefit from a reduced number of Krylov iterations. For stiffer equations such

as Eq. (3.15), fill levels of f = 4 to f = 6, depending on the problem size, have been shown

to improve convergence rates of iterative Krylov solvers significantly. For elliptic PDEs such

as Eq. (3.15), methods such as multigrid, which is introduced in the next section, are a far

superior choice to preconditioned iterative Krylov approaches.

Multi-Grid

In this section, a brief background on multi-grid methods is given. In essence, the multi-grid

method works by transferring a problem of interest onto increasingly coarse grids, where the

solution is approximated, which in effect filters out high-frequency errors in the solution at each

coarsening step. A simple representation of the multi-grid algorithm is provided in Algorithm

44

3.2, and an illustration of the transfer of solutions to increasingly coarse grids is given in Fig.

3.6.

Algorithm 3.2: V-Cycle multi-grid algorithm for solving Ax = b.

function multigrid(Ak, b, u, k)if k < N then

u = S1m(Ak, b, u)

Rk = compute restrictor(Ak)Pk = compute interpolator(Ak)rk+1 = Rk(b−Aku)Ak+1 = RkAkPkek+1 = 0ek+1 = multigrid(Ak+1, rk+1, ek+1, k + 1)u = u + Pkek+1

u = S2m(Ak, b, u)

elseu = direct solve(Ak, b)

endreturn u

end function

In Algorithm 3.2, u is the solution vector, k denotes the coarse grid level, Rk and Pk are

the restriction and prolongation matrices respectively, rk is the residual vector, ek is the error

vector, and S1m and S2

m denote the pre- and post-smoothers respectively.

In general, the choice of S1m and S2

m is problem dependent, and is the primary factor in

the efficiency of multi-grid methods. For scalar problems, methods such as Successive Over-

Relaxation (SOR) or Incomplete Lower-Upper (ILU) factorization work relatively well and are

good choices for the systems of equations arising from segregated solvers such as the fractional-

step projection method. However, for schemes such as coupled schemes, iterative schemes may

not be sufficient. In fact, it has been known for decades that an appropriate multi-grid method

applied to a linear system arising from the discretization of a positive definite elliptic PDE,

which includes Eq. (3.15), will represent an optimal solver, with a time complexity of O(n),

compared to O(n2) for an iterative Krylov solver. There are two main variations of the multi-

grid algorithm; geometric and algebraic. The difference between the two manifests itself in the

computation of the restrictor and prolongation operators, Rk and Pk. In a geometric multi-grid,

these operators are constructed using the connectivity information from the grid on which the

problem is discretized. For algebraic multi-grid, these operators must be constructed from the

45

k = 0

k = 1

k = 2

R0

R1

P1

P0

Figure 3.6: Illustration of the v-cycle multi-grid coarse grid levels. Rk denotes the applicablerestriction operator and Pk is the prolongation operator.

coefficient matrix itself, for example by using the sparse matrix graph and linear combinations

of fine grid unknowns.

Implementation of the Linear Solver

The software developed in this work is compatible with several different numerical libraries.

Serial direct solvers and Krylov iterative solvers are available through the Eigen library [109].

Parallel direct solvers, which are needed for the coarse multigrid direct solve are implemented

using the Amesos2 package [110] which is distributed as part of the Trilinos library [111]. A

variety of parallel Krylov subspace solvers are also available from the Belos package [110],

including an implementation of the BiCGSTAB solver. Schwarz global preconditioning and

the ILU(f) local preconditioner are implemented using Ifpack2 [112]. The geometric multigrid

solver is incorporated through the MueLu library [113]. In general, BiCGSTAB iterations in

conjuction with an additive Schwarz/ILU(f) preconditioner is used to solve well-conditioned

46

linear systems, such as those that arise from the discretization of time-dependent equations.

For linear systems that arise from the discretization of elliptic PDEs, a Krylov method such

as BiCGSTAB, GMRES or Transpose-Free Quasi-Minimal Residual (TFQMR) is used, with a

geometric multigrid solver applied as a global preconditioner.

3.7.2 Parallel Implementation

In order to facilitate faster computation, the domain is decomposed into subdomains. The Mes-

sage Passing Interface (MPI) is used to facilitate communication between separate subdomains.

Each subdomain contains a region which overlaps the neighbouring subdomain. In a typical

FVM discretization, the amount of overlap of the individual subdmains is dictated by the order

of the reconstruction used to compute the solution variables on cell faces. In an IBM context,

the number of cells from a neighbouring processor needed to apply the boundary conditions

may not be known in advance. In general, the overlap of the subdomains is therefore dictacted

by a fixed width rather than by a fixed number of cell neighbours. In this work, the domain is

decomposed in such a way as to limit the number of edge cuts in the connectivity graph of the

FVM grid using the METIS library [114]. The size of the overlapping regions is set by a fixed

radius. Fig. 3.7 depicts a computational grid divided up into subdomains, as well as a single

local subdomain with buffers used to facilitate inter-process communication.

0

1

2

34

5

6

7

5 72

4

Figure 3.7: The partitions generated by the domain decomposition scheme.

After a solution variable is updated, it is necessary to synchronize the domain across pro-

cesses. This is accomplished through the use of non-blocking MPI calls. After the local values

have been computed, each subdomain updates the overlapping cells which are not owned lo-

47

cally, so that each process has the relevant solution data before proceeding to the next step in

the computation.

48

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61

Chapter 4

A volume-of-fluid ghost-cell

immersed boundary method for

multiphase flows with contact line

dynamics

With minor changes to fulfill formatting requirements and consistency with the symbology in the rest ofthis work, this chapter is largely as it appears in A. O’Brien and M. Bussmann, “A Volume-of-Fluid Ghost-Cell Immersed Boundary Method for Multiphase Flows with Contact Line Dynamics,” Computers and Fluids,vol. 165, pp. 43–53, Mar. 2018. doi: 10.1016/J.COMPFLUID.2018.01.006.

62

4.1 Abstract

A method is proposed for modelling contact line dynamics in volume-of-fluid frameworks using

a common and robust ghost-cell immersed boundary method in 2D. The method implements

a contact line model based on the CELESTE method, used in conjunction with a CICSAM

scheme capable of handling multiphase flows in which the phases have large variations in their

physical properties. The ability of the model to resolve contact lines on both planar and curved

geometries is demonstrated. It is also shown that the model maintains good mass conservation.

Dynamic cases studied include a planar water droplet impact onto a cylinder, as well as a planar

droplet penetration into a bank of cylinders with varying hydrophobicity.

4.2 Introduction

Many flows, both industrial and natural, involve the wetting/de-wetting of a solid body, in which

one fluid is displaced by another. The two fluids often have very different physical properties

and are immiscible, resulting in the formation of a nearly discontinuous interface. At the point

of contact on the solid body, the interface intersects the body at an angle dependent on the

surface energies of the two fluids and the solid, governed by the well-known Young’s equation:

θs = arccos

(γs,l1 − γs,l2

γl1,l2

)(4.1)

where θs is the static contact angle, γ is a surface energy and the subscripts s, l1 and l2 denote

the solid and fluid phases. Many engineering applications must carefully consider interface

dynamics, a few of which include: separation of oil from solid particles [2], ink-jet printing

[3], drug delivery through nano- and micro-scale capsules [4], and the cooling of microchan-

nels [5]. In this work, we present a new algorithm for simulating contact angle dynamics on

arbitrary bodies using Computational Fluid Dynamics (CFD). In particular, a method is pro-

posed that aims to combine Volume-of-Fluid (VoF) with sharp-interface Immersed Boundary

Methods (IBMs). Below a brief summary of IBMs is presented, followed by a discussion of the

numerical components required to model multiphase flows, as well as some recent works that

have attempted to reconcile the two fields.

63

The Immersed Boundary Method (IBM) [6, 7] involves using a fixed Eulerian background

mesh on which the fluid equations of motion are solved, with solid bodies represented as La-

grangian entities. Since the mesh does not conform to the shape of the body, forces imparted

by the body on the fluid (and vice versa) must be accounted for through a coupling term in

the momentum equations. IBMs were pioneered by Peskin for use in modelling cardiac flows

[6], and have since been adopted and extended for many different applications, including fluid-

structure interaction [8], flow over flexible membranes [9], formation of ice on an airfoil [10],

and ship hydrodynamics [11]. The original IB method of Peskin involves introducing a spring-

like continuous forcing function into the momentum equations, and thus belongs to the first

category of IB methods known as Continuous Forcing (CF) methods. Due to the discrete na-

ture of the background Eulerian mesh, CF methods require the forcing to be spread to several

nearby cells through a spreading function. A second category of IB methods known as Discrete

Forcing (DF) methods also exists, where the forcing is introduced into the discretized momen-

tum equations. This method produces a sharper interface between the fluid and solid regions,

since only cells with immediate proximity to the IB are forced. This second category therefore

enjoys a significant advantage in the context of rigid body simulations, and is adopted here. An

extensive review of various IBMs in both categories was published by Mittal and Iccarino [7].

Body-fitted meshes have long been used for the simulation of contact lines on surfaces of

various shapes [12–16]. An excellent review of contact angle methods for body-fitted meshes is

presented by Sui et al. [17]. The use of IBMs in lieu of body-fitted meshes is attractive primarily

for the flexibility/ease in handling arbitrary geometries and the potential for modelling systems

of solid bodies with arbitrary relative motion. While IBMs have been applied previously in the

context of multiphase flows, usually for the study of wave-structure interaction [11, 18–23], the

effect of either surface tension or wetting dynamics is usually negligible and often neglected.

For other classes of flow problems, such as wetting/de-wetting of small particles [2] and flow

through porous media, surface tension forces may comprise the dominant momentum term.

A few methods have been proposed for combining wetting dynamics and IBMs. Yang and

Stern [18] combined a sharp-interface IBM with a level-set method, and developed a contact

angle model for studying wave-body interactions with surface tension. They noted that the

modification of the level-set function led to issues with phase conservation. This was also

64

observed by Liu and Ding [24], who proposed a diffuse-interface method for Moving Contact

Line (MCL) problems.

Two primary capabilities are required for the successful simulation of multiphase flows with

surface tension: (i) advection of a colour function representing the distribution of phases, and

(ii) accurate calculation of surface tension forces from the colour function. Tryggvason [25] et

al. proposed the Front-Tracking (FT) method which represents the interface as a Lagrangian

entity represented by a set of marker points. The Level-Set (LS) method (e.g. Sussmann

et al. [26]) describes the distribution of phases in terms of a signed distance function, with

zero representing the interface. This method is simple and has several advantages, including

minimizing spurious interface velocities, but suffers from a lack of mass conservation. The

Volume-of-Fluid (VoF) method of Hirt and Nichols [27], represents the distribution of phases

in each cell as volume fractions. This method conserves mass exactly, but is somewhat more

difficult to advect without significant numerical diffusion. Several schemes to advect the VoF

field and maintain a sharp interface between phases exist, including geometric schemes such

as Simple Linear Interface Calculation SLIC [28] and Piecewise Linear Interface Calculation

PLIC [29]. While geometric schemes are accurate, they are difficult to couple with IB methods,

particularly when the surface of the IB is curved, since in principle the interface would need to

be constructed only in the portion of the cell that is not occupied by the IB. An alternative

to geometric interface advection schemes are so-called compressive interface advection schemes,

such as the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) of Ubbink

and Issa [30]. In compressive interface advection schemes, a downwind component is introduced

to the advection scheme, and the Normalized Variable Diagram (NVD) is used to determine

the weights of the upwind and downwind components such that the Convective Boundedness

Criterion (CBC) is satisfied. This approach can be readily implemented in any IBM framework,

since no underlying assumptions about the cell shape are required.

The imposition of surface tension forces on the momentum equation requires the calculation

of interface normals and mean curvature. While the evaluation of the interface normals is

relatively straight-forward so long as the colour function gradient is available, the estimation

of the interface mean curvature is much more challenging. This can in principle be done in

a relatively straight-forward manner on regular cartesian grids by using the height function

65

method [27] in conjunction with a contact line model such as the one developed by Afkhami

et al. [31]. An issue arises, however, in that it may not be practical to sufficiently extend the

VoF field into the IB such that well-defined height functions exist. Instead, a method that has

proven to work well with irregular stencils should be chosen. An alternative approach is to use

a least-squares type approach, such as the CELESTE method proposed by Denner and van

Wachem [32]. This method was originally proposed for irregular unstructured grids, but can

be modified to accommodate other cases where a modified stencil is necessary, as will be shown

in this paper.

In this paper, a new method is presented for enforcing contact angle dynamics on arbitrarily

shaped IBMs on 2D cartesian grids. A volume-of-fluid (VoF) method is used to distinguish the

two fluid phases, and a sharp-interface IBM based on the ghost-cell method of Majumdar

et al. [33] is implemented. In order to incorporate surface tension and wetting dynamics, a

modified version of the CELESTE method [32] is implemented. The ability of the algorithm to

enforce wetting conditions on both planar and curved geometries is demonstrated.

4.3 Numerical Methodology

4.3.1 Governing Equations

Unsteady, incompressible viscous flows are governed by the Navier-Stokes equations. For vari-

able density, incompressible flows these consist of the momentum equation,

∂ρu

∂t+∇ · ρuu + p

~I =∇ ·

(µ[∇u + ()∇uT

])+~fb (4.2)

and the divergence free velocity field condition,

∇ · u = 0 (4.3)

where ρ is the density, u is the velocity, p is the pressure, µ is the viscosity,~I is the identity

tensor and ~fb are all other body forces, including gravity and surface tension. In volume-of-fluid

methods [27], a colour function is transported which represents the volume fraction of one phase

with respect to the total cell volume. This equation takes the form

66

∂γ

∂t+∇ · uγ = 0 (4.4)

In VoF methods, only a single set of equations is solved, with ρ and µ computed using γ as

ρ(γ) = (1− γ)ρ1 + γρ2 (4.5)

µ(γ) = (1− γ)µ1 + γµ2 (4.6)

Since ρ = ρ(γ), Eq. (4.4) also serves as the continuity or mass conservation equation. Finally,

the body forces ~fb in Eq. (4.2) consist of the surface tension and gravitational forces,

~fb = σκδm + ρg (4.7)

where σ is the surface tension coefficient, κ is the interface curvature, δ is the Dirac delta

function, m is the interface normal vector and g is the gravitational acceleration.

4.3.2 Discretization of the Navier-Stokes Equations

Eqs. (4.2) and (4.3) in this work is discretized by a second order fractional-step method, with

body forces and pressure discretized using the face-to-cell approach proposed by Francois et

al. [34]. Gravity is included by transforming the gravitational body force into an interface force

similar to the approach of Montazeri et al. [35], allowing for ~fb to be expressed as ~fb = ~fb(γ).

Numerical Discretization

In this section, the discretization of the Navier-Stokes equations is briefly presented. A three-

step fractional-step method is used. First, a predicted velocity u∗ is solved by discretizing Eq.

(4.2), resulting in

ρn+1c u∗c − ρncunc

∆tΩc +

∑f

Fnmu∗f ·~sf =∑f

µnf∇u∗f ·~sf − Ωcρn+1c

∇pnf −~fnb

ρnf

f→c

(4.8)

67

where Ωc is the cell volume, f → c denotes a face-to-cell interpolation, and the superscript

n denotes a previous time level. Face gradients are discretized using the generalized scheme

of Mathur and Murthy [36], and body forces on the right-hand side are interpolated to the

cell centres using the method proposed by Francois et al. [34]. The momentum flux Fnm must

be implemented in a manner consistent with the method used to discretize Eq. (4.4) as noted

by Bussmann et al. [37], and will be discussed further in a following section. In order to

transform the gravitational source term ρg in Eq. (4.7) into an interface term, we first employ

the substitution

p = ps + ρg ·~r (4.9)

where ps is the classical interpretation of pressure. Evaluating the gradient we get

∇p =∇ps +∇ρg ·~r + ρg (4.10)

and therefore

∇p−∇ρg ·~r =∇ps + ρg (4.11)

Since ρ = ρ(γ), gravity can be expressed in terms of γ and a density jump ∆ρ = ρ2 − ρ1:

∇ps + ρg =∇p−∆ρg ·~r∇γ (4.12)

Employing the CSF method of Brackbill et al. [38] for the discretization of the surface tension

force, the full body source term can be expressed as a single interface force of the form

~fnb = σκ−∆ρg ·~r∇γ (4.13)

In the second step we remove ∇pn from u∗c and compute u∗∗. Note that the point of

retaining this term in Eq. (4.8) was to ensure valid boundary conditions were used to compute

u∗ to ensure a correct projection near boundaries, as discussed in detail by Brown et al. [39].

Thus u∗∗ becomes

68

u∗∗c = u∗c + ∆t

∇pnfρnf

f→c

(4.14)

u∗∗f = u∗ + ∆t

∇pnf −~fnb

ρnf

f→c

+ ∆t~fn+1

b

ρn+1f

(4.15)

where the bar denotes a quantity that is interpolated using a simple arithmetic mean.

The third step corrects u∗∗ to be divergence free. A Hodge decomposition of u∗∗ leads to a

pressure Poisson equation of the form

∆t∇ · ∇pn+1

ρn+1=∇ · u∗∗ (4.16)

which when discretized becomes

∆t∑f

∇pn+1f

ρn+1f

·~sf =∑f

u∗∗f ·~sf (4.17)

Finally, the cell and face velocities are corrected using

un+1c = u∗∗c −∆t

∇pn+1

f

ρn+1f

f→c

+ ∆t

~fn+1

b

ρn+1f

−~fnbρnf

f→c

(4.18)

un+1f = u∗∗f −∆t

∇pn+1f

ρn+1f

(4.19)

4.3.3 Discretization of the VoF Equation and Momentum Flux

In the ghost-cell IB method, exact truncated cell geometries are not used for the finite-volume

discretization, and thus using a geometrical interface advection scheme such as Piecewise Linear

Interface Construction (PLIC) [40–44] is not feasible. Here, the CICSAM scheme of Ubbink

and Issa [30] is used instead, with careful consideration to ensure the resulting mass fluxes

obtained from Eq. (4.4) are exactly consistent with the momentum flux Fm used to advect u∗,

which as noted by Bussmann et al. [37] is critical for flows with significant density ratios. In

this section we describe the method used to ensure consistency between the discretizations of

Eq. (4.2) and Eq. (4.4). First, the value of γ for a donor cell is normalized according to the

69

Normalized Variable Diagram (NVD) of Leonard [45]

γD =γD − γUγA − γU

(4.20)

where γU , γD and γA are the values of γ at the upwind, donor and acceptor cells respectively.

As proposed by Jasak et al. [46], the value of γU can be computed generally as

γU = γA + 2xD − xA ·∇γD (4.21)

where xD and xA are the position vectors of the donor and acceptor cells respectively. The two

schemes used to compute the normalized value of γ at the face are the Hyper-C (HC) scheme:

γHCf =

min 1, γD

CoDwhen 0 ≤ γD ≤ 1

γD otherwise

(4.22)

and the Ultimate Quickest (UQ) scheme [45]:

γUQf =

min 8CoD γD+(1−CoD)(6γD+3)

8 , γHCf when 0 ≤ γD ≤ 1

γD otherwise

(4.23)

The HC scheme has a tendency to artificially distort the interface as it moves tangentially to cell

faces, while the UQ scheme is overly dissipative. In order to remedy this, a blending function

is employed to combine both schemes, such that

γf = ψf γHCf + (1− ψf )γUQf (4.24)

where ψf is a weighting function that depends on the interface orientation. Ubbink and Issa

[30] proposed the following form for ψf :

ψf = min kψcos (2θf ) + 1

2, 1 (4.25)

where the angle θf is defined as

70

θf = arccos

∥∥∥∥m · ~rc|~rc|∥∥∥∥ (4.26)

The constant kψ adjusts the relative dominance of Eqs. (4.22) and (4.25). Ubbink and Issa [30]

proposed a value of kψ = 1, which is used in this work. Once γf is determined from Eq. (4.24),

it is used to compute a weighted linear interpolation parameter β from the equation

β =γf − γD1− γD

(4.27)

and the colour function value γf is interpolated as

γf = (1− β)γD + βγA (4.28)

which is then used as the face centered value in the discretized form of Eq. (4.4). Additionally,

the term Fm in Eq. (4.8) should be computed as

Fm = (1− β)γD∆ρ+ βγA∆ρ+ ρ1unf (4.29)

Note that Eq. (4.29) effectively means a different value of ρf is used for advecting momentum

than is used in Eq. (4.17). Using this interpolation method alongside the density weighted face-

to-centre evaluation of the pressure gradient described by Francois et al. [34] has enabled us to

simulate flows of density ratios up to 1012 without need for the use of harmonic interpolation

of densities at faces, or computing the densities based on the convolved γ field as suggested in

[47]. Upon computing the flux term, a simple upwind scheme is used to compute u∗.

4.3.4 Computing the Surface Tension Force

In the CSF method of Brackbill et al. [38], the interface force is ~fi = σκ∇γ, with the curvature

κ computed as

κc =∇ · m (4.30)

where m is the interface unit normal vector, defined as

71

m = − ∇γ‖∇γ‖

(4.31)

where γ is a convolved volume fraction field, explained in the following section.

Computation of the Interface Normals

The unit normal vector to the interface is computed directly from a smoothed γ field. Using the

actual γ field leads to large errors due to the discontinuous nature of the interface. Numerous

authors [38, 48] have noted that these errors can be greatly reduced by computing the gradient

based on a convolved γ field instead, denoted as γ. Thus, for the purpose of evaluating the

interface normals, a convolved volume fraction field γ is computed as

γ =

∫ΩK(x)γdΩ (4.32)

where the kernel function K(x) is a simplified version of the one used by Peskin [6], which takes

the form

K(x′) =

A

2rk1 + cos(πr/rk) if r < rk

0 otherwise

(4.33)

where rk is the effective radius of the kernel, A is a constant specified by the unity property

and r = ‖x′ − x‖. Through experimentation, it was found that a kernel radius of rk = 2∆l,

where ∆l is a characteristic cell length, resulted in the smallest parasitic currents. In order to

compute ∇γ in the CELESTE scheme [32], the Taylor series of γ is expanded to second-order

from neighbouring cells to the cell centre of interest in the following manner:

γc = γn +∂γn∂x

∆x+∂γn∂y

∆y +∂2γn∂x2

∆x2

2+∂2γn∂y2

∆y2

2+∂2γn∂x∂y

∆x∆y +O(∆x3,∆y3) (4.34)

Eq. (4.34) can be arranged into a weighted, over-determined linear system of the form Ax = b,

in terms of the unknown derivatives, yielding

72

∆x1∆s21

∆y1∆s21

∆x212∆s21

∆y212∆s21

∆x1∆y1∆s21

......

......

...

∆xi∆s2i

∆yi∆s2i

∆x2i2∆s2i

∆y2i2∆s2i

∆xi∆yi∆s2i

......

......

...

∆xN∆s2N

∆yN∆s2N

∆x2N2∆s2N

∆y2N2∆s2N

∆xN∆yN∆s2N

∂γ∂x

∂γ∂y

∂2γ∂x2

∂2γ∂y2

∂2γ∂x∂y

=

∆γcn1

∆s21...

∆γcni

∆s2i...

∆γcnN

∆s2N

(4.35)

Haselbacher and Vasilyev [49] identified two constraints on the use of least-squares methods

for reconstructing a gradient from a Taylor series: (1) the stencil should be as small as possible

to reduce truncation error, and (2) the stencil should be as symmetric as possible in order to

cancel out truncation error terms. Since at least five cells are needed to solve Eq. (4.35), the

cell face and diagonal neighbours are used, resulting in the 3×3 Cartesian stencil shown in Fig.

4.1.

(a)

Contact line

Interface

mcl

(b)

Figure 4.1: Least-squares stencil used for both the evaluation of m and κ, for (a) a regular celland (b) boundary cell.

As per Fig. 4.1 (b), when adjacent cell neighbours are not available, for example along bound-

aries, the corresponding face or node is used instead, with the value of γ depending on the

boundary condition.

73

Computation of Interface Curvature

The procedure for computing the curvature follows a similar process to the interface normals.

Eq. (4.30) is expanded as

κ =∇ · m =∂nx∂x

+∂ny∂y

(4.36)

A linear system is then solved for each component of m, denoted by the subscript j, taking the

form

∆x1 ∆y1∆x21

2∆y21

2 ∆x1∆y1

......

......

...

∆xi ∆yi∆x2i

2∆y2i

2 ∆xi∆yi...

......

......

∆xN ∆yN∆x2N

2∆y2N

2 ∆xN∆yN

∂nj

∂x

∂nj

∂y

∂2nj

∂x2

∂2nj

∂y2

∂2nj

∂x∂y

=

∆mj,cn1

...

∆mj,cni

...

∆mj,cnN

(4.37)

where nj is a component of the unit normal vector m. Our numerical experiments have

shown consistently that more accurate curvatures are achieved when using an un-weighted

least-squares procedure for the computation of interface normal divergence. Upon solving Eq.

(4.37), the appropriate derivatives are substituted into Eq. (4.36) to obtain the cell centered

curvature. This approach differs slightly from the original approach of Denner and van Wachem

[32], in that for the evaluation of κ, they considered only the first derivatives of m were consid-

ered. Overall, the use of higher order derivatives was found to produce a smoother curvature

field and fewer parasitic currents for the cases studied here.

Finally, unlike the original CELESTE method, no smoothing of the curvature was per-

formed. Instead, the curvature was simply not evaluated in cells where ‖∇γc‖ < ε, where ε is

a small cut-off value. For interpolating curvatures to faces, ‖∇γf‖ < ε was also enforced, even

if the cell centred gradient was non-zero.

4.3.5 Ghost-Cell Immersed Boundary Method

In recent years, immersed boundaries have received much attention for simulating flows over

complex bodies [7]. The “ghost-cell” IB method is a direct forcing method that defines the flow

74

variables within the boundary itself, such that the effect of the boundary is incorporated when

flow variables are computed on the cell faces. Our ghost-cell method follows that of Tseng and

Ferziger [50] with a slight modification for near-boundary cells. In Tseng and Ferziger’s method

[50], a symmetric interpolation stencil is constructed across the boundary as shown in Fig. 4.2.

IB cell Fluid cell

Stencil A Stencil B

xbpB

xipB

xbpA

xipA

Figure 4.2: Interpolation stencil used in the ghost-cell immersed boundary method, showingthe image point xip and boundary point xbp.

The nearest intersection point from the IB cell is found, and an image point in the fluid domain

is computed as

xip = 2xbp − xgc (4.38)

The nearest grid node to the image point is then found, and the four cells that connect to this

grid node are used for bilinear interpolation. The bilinear polynomial for an arbitrary scalar φ

is defined as

φ = a3xy + a2x+ a1y + a0 (4.39)

which can be used to compute the value of φ within the bounded region. Thus, the four

coefficients a0 − a3 in Eq. (4.39) must be evaluated. In order to compute these coefficients, a

Vandermonde matrix [51] is constructed using the four bounding cell centers shown in Fig. 4.2

75

1 y1 x1 x1y1

1 y2 x2 x2y2

1 y3 x3 x3y3

1 y4 x4 x4y4

a0

a1

a2

a3

=

φ1

φ2

φ3

φ4

(4.40)

In order to implicitly evaluate Eq. (4.40) when solving Eq. (4.8), the coefficient matrix is

inverted to obtain the unknown coefficients in terms of a linear combination of the values of

φ at the bounding cell centers. It can easily be shown that an un-modified ghost-cell method

can break down if the ghost-cell is close to the IB and is thus part of the image point stencil.

In order to avoid this, we use a different approach when the image point stencil includes the

ghost-cell. For stencil A in Fig. 4.2, the boundary point is used instead of the image point,

resulting in the following linear equations for Neumann and Dirichlet conditions:

Dirichlet:∑s

aiφi =

xbpybp

xbp

ybp

1

T

V −1φ (4.41)

Neumann:∑s

aiφi =

ybpnx + xbpny

nx

ny

0

T

V −1φ (4.42)

where∑

s denotes a sum over all the stencil cells, V and φ are the Vandermonde and right-

hand-side vectors in Eq. (4.40) respectively, and nx and ny denote the x and y components of

the unit normal vector of the IB at the boundary point. The above procedure works well when

the ghost-cell is close to the IB, but can break down as the IB approaches one of the other

image point cells. Therefore, in the case of stencil B, a standard ghost-cell method is used, with

76

the image point value computed as

φip =

xipyip

xip

yip

1

T

1 y1 x1 x1y1

1 y2 x2 x2y2

1 y3 x3 x3y3

1 y4 x4 x4y4

−1

φ1

φ2

φ3

φ4

(4.43)

The Dirichlet and Neumann conditions can then be evaluated along the symmetric stencil

as follows:

Dirichlet:φip + φgc

2= φbp (4.44)

Neumann:φgc − φip

∆l= φbp (4.45)

where ∆l is the scalar length of the interpolation stencil. Finally, if Robin boundary conditions

are desired, a linear combination of the of Dirichlet and Neumann conditions proposed above

can be used.

Degenerate Cases

Degenerate cases can make the construction of an IB stencil ambiguous. For example, an IB cell

centre may lie at an equal distance from two edges of a polygonal IB, such as the case depicted in

Fig. 4.3. Cases such as this can lead to significant issues in simulations where mass conservation

is important. In order to minimize the impact of such degenerate cases, two possible remedies

exist. One approach is to overlay a second cell on top of the IB cell, effectively allowing for two

stencils to be constructed, but introducing an extra unknown into the problem. While effective

and conceptually simple, its implementation involves a lot of programming overhead since the

cell connectivity has been altered. Instead, the approach applied here is when more than one

boundary point candidate is found within some tolerance, the nearest IB vertex is used instead.

77

xbp,1

xip,1

xbp

xip

xbp,2xip,2

Figure 4.3: Degenerate case resulting in an ambiguous choice of interpolation stencil. The redlines show the two possible stencils based on a nearest intersection, while the black stencil isthe actual stencil used.

4.3.6 Implementing Contact Line Dynamics on Ghost-Cell Immersed Bound-

aries

Although IB methods have seen increasing attention over the years due to their versatility,

efficiency and increasing accuracy, little work has been done on the development of multiphase

models coupled to IB methods. Liu and Ding [24] developed a moving contact line model

for the Direct Forcing (DF) IB method of Ren et al. [52], using a diffuse-interface multiphase

model. Zhang et al. [20], and Fotis and Yang [8] used an IB model to simulate wave-structure

interactions at free surfaces, but did not consider surface tension effects. Niels et al. [53] used

IB methods for multiphase flows, but did not consider contact line dynamics. Ghasemi et

al. [22] studied interactions between moving solids and two-phase systems using a PLIC-VOF

multiphase method, but also did not consider contact line dynamics. Here we propose a simple

method for modelling contact angle dynamics based on the CELESTE algorithm of Denner and

van Wachem [32], which is effective even for curved surfaces.

Computing the Interface Curvature Near the Immersed Boundary

In order to compute the interface curvature in cells neighbouring an IB cell, we employ a

straight-forward method involving a modified CELESTE stencil. The orientation of the inter-

face normal at the contact line, mcl, where the stencil intersects the IB is known, since it can

78

be determined using the prescribed contact angle. Thus, the CELESTE stencil in Fig. 4.1 is

modified near the IB as shown in Fig. 4.4.

Interface Contact line

θ~ncl

Figure 4.4: Least-squares stencil used for the evaluation of κ near the IB surface.

At each point where the stencil intersects the IB, the value of mcl is used instead of a

value of m calculated from Eq. (4.31) for the calculation in Eq. (4.37). Therefore, cells which

neighbour the IB account for the proximity of the IB when computing κ, creating a wetting force

in Eq. (4.2) when the interface intersects the IB at an angle that deviates from the prescribed

contact angle. For the computation of ∇γ, an unmodified CELESTE stencil which includes

the ghost-cells is used.

4.4 Results

A number of test cases are presented to demonstrate that the proposed numerical scheme

enforces the correct wetting dynamics along the surface of an immersed boundary. The first

case simply demonstrates the ability of the proposed algorithm to enforce the contact angle on a

curved surface. Then, we investigate the quality of the results for both well-aligned and poorly

aligned IBs. We then investigate two applications, a planar droplet impact onto a cylinder and

a planar droplet penetration into a porous media. For all cases, the time-step is constrained by

imposing a maximum allowable CFL number of 0.25, a condition identified by Ubbink and Issa

79

[30] as necessary for maintaining a sharp interface when using a CICSAM advection scheme,

as well as by the capillary time-step constraint identified by Brackbill et al. [38] defined as

∆tσ ≤√

(ρ1+ρ2)∆l3

4πσ where ∆l is a characteristic cell length scale defined as min ‖~rc‖.

4.4.1 Imposing a Static Contact Angle on an IB Cylinder

In this test case, the ability of the proposed method to maintain a contact angle on a cylindrical

geometry is demonstrated. The cylinder has a radius rs = 0.5mm, and the droplet is initialized

with a radius of curvature rl,0 = 0.5mm, and is initially spaced such that the interface intersects

the cylinder at θ = 90. Fig. 4.5 depicts the configuration of the computational domain.

dc,0

α

θ

rl,0

rc

4rc

Figure 4.5: Initial configuration for a planar droplet on a horizontal cylinder.

In the absence of gravity, the equilibrium configuration can be computed analytically. The area

of the planar droplet can be computed as

A0 = πr2l,0 − r2

c arccos

d2c,0 + r2

c − r2l,0

2dc,0rc

− r2

l,0 arccos

d2c,0 + r2

l,0 − r2c

2dc,0rl,0

+

√(−dc,0 + rc + rl,0)(dc,0 + rc − rl,0)(dc,0 − rc + rl,0)(dc,0 + rc + rl,0)

4

(4.46)

80

where rl,0 is the initial radius of curvature, rc is the radius of the cylinder, and dc,0 is the

initial spacing between the centres of the cylinder and the droplet. The final configuration of

the droplet can then be found in terms of the spacing dc and droplet radius of curvature rl by

finding the roots of the equation

f(dc) = A0 − (α+ θ)r2l + αr2

c − rlrc sin θ (4.47)

where

α(dc) = arccos

r2c + d2

c − r2l

2rcdc

(4.48)

rl(dc) = rc cos θ + 2√r2c cos θ2 − r2

c + d2c (4.49)

Five different contact angles were tested to demonstrate the ability of the numerical scheme

to handle a wide range of wettability, on an equidistant cartesian mesh with a resolution of

225× 300. This results in 150 cells across the horizontal diameter of the cylinder. The contact

angles tested were 30, 60, 90, 120 and 150. In each case the initial interface is configured

such that it intersects the IB surface at 90, and the simulation is allowed to run until the

velocity residuals decay to a reasonably small value. The final contact angle was measured by

fitting a circle to the γ = 0.5 contour, and computing its intersection angle with the cylinder.

Fig. 4.6 depicts the final numerically obtained interface configurations at varying contact angles.

The hydrophilic cases produced the largest errors in phase conservation. This is likely

a result of the large area projection of the planar droplet onto the IB [54], as well as the

higher overall spreading rate. The sharp-interface ghost-cell immersed boundary method has

an advantage in that when the IB aligns well with adjacent cell faces, the boundary condition

reduces to the face value as if an actual boundary were present. When the mesh is not well-

aligned, an error in phase conservation is introduced. Fig. 4.8 shows the trend of mass error,

as a percentage loss/gain in total volume of the immersed phase, as a function of mesh spacing

h for the 30 case.

The mass loss itself is due to the well-known issue of transpiration error in IB methods. The

81

(a) θ = 30 (b) θ = 90 (c) θ = 120

Figure 4.6: Static contact angles imposed on a cylindrical IB.

(degrees)

r l/rs

30 60 90 120 1500.8

1

1.2

1.4

1.6

Exact

Numerical

Figure 4.7: Numerically computed droplet radii at various contact angles.

mass conservation can be quantified by considering the phase conservation equation for an IB

cell at steady state:

dγ

dt= −

∫Sb

ufγf · ~ndS (4.50)

where Sb denotes the IB surface. When an IB cell is discretized on an equidistant cartesian

mesh, and assuming a no-slip boundary condition for velocity, and a Neumann condition for γ,

82

h

Em (

%)

0.010.020.03

0.2

0.4

0.6

0.8

2nd

Order

Figure 4.8: Mass loss error per unit non-dimensional time for the planar droplet situated on acylinder with θ = 30. Note that for this case, t∗max = 0.2.

this equation becomes

dγ

dt= −

∑f

1

4up − uip · γp + γip (4.51)

which approaches zero if the solution is at steady state, or in the infinitesimal limit where

φp = φip. As seen in Fig. 4.8, the order of convergence with respect to mass error is consistent

with the known order of accuracy of the bilinear interpolation used in the ghost-cell method. A

detailed summary of all the known mass conservation errors associated with the ghost-cell IB

method can be found in Seo and Mittal [55]. Finally, the convergence of the contact angle was

tested for five different mesh resolutions. The grid convergence of the θ = 30 case is depicted

in Fig. 4.9. The method converges rapidly with mesh refinement, with the solution on the three

finest meshes being nearly indistinguishable.

4.4.2 Imposing a Contact Angle on a Poorly Mesh-Aligned Plane

In this section, initially semi-circular planar droplets are placed on rectangular IBs, which are

oriented at either 0 or 45. The first represents an optimal alignment of the IB with respect to

the background mesh, while the latter represents a worst case alignment. No gravity is present,

83

105×140

75×100

150×200

210×280

Anaylitical

300×400

Figure 4.9: Convergence of the proposed method with grid refinement, with θ = 30.

thus only the wetting force drives the droplet to spread or contract. The Ohnesorge number for

all cases is Oh = 0.1. Figs. 4.10 and 4.11 depict the obtained solutions for the flat and tilted

platforms respectively.

(a) θ = 45 (b) θ = 90 (c) θ = 135

Figure 4.10: Planar droplets situated on a flat, well-aligned IB. The bottom droplets are placedon a regular domain boundary for reference.

Additionally, it can easily be shown that the internal pressure for a planar droplet in zero

gravity as a function of contact angle and area is:

p = σ

√2θ − sin (2θ)

2A(4.52)

84

(a) θ = 45 (b) θ = 90 (c) θ = 135

Figure 4.11: Planar droplets situated on a tilted, poorly-aligned IB. The bottom droplets areplaced on a regular domain boundary for reference.

Fig. 4.12 depicts the numerically computed internal pressure for both the flat and tilted planes

at various contact angles.

θ (degrees)

∆P

R0/σ

0 30 60 90 120 150 1800

0.05

0.1

0.15

0.2

0.25

0.3

Analytical

Flat Platform

Tilted Platform

Figure 4.12: Internal droplet pressure at varying contact angles.

A difference in the droplet internal pressure is observed between the flat and tilted platforms

at small contact angles. The difference in pressure between the hydrophilic and hydrophobic

platforms is due to greater error in extrapolating the γ field to the ghost-cells from the fluid

domain at small contact angles, resulting in greater errors in the computation of∇γ near where

the fluid-fluid interface intersects the IB. This issue becomes most pronounced when the mesh

85

alignment is poor, resulting in a larger IB stencil length.

Finally, in order to validate the wetting force on the platform, the spreading rate was

compared to the well-known Tanner law [56]. For viscous flows, droplet spreading can be

characterized into two regimes; an early inertial spreading phase in which the non-dimensional

droplet radius changes as R(t) ∝ t1/2, and a viscous spreading phase in which R(t) ∝ t1/10.

t

(R/R

0­1)/Oh

0.2 0.4

10

20

30

40

Oh=0.05

Oh=0.1

Oh=0.25

R(t) ∝ t1/10

R(t) ∝ t1/2

Figure 4.13: Non-dimensional droplet radius with respect to time for a planar droplet on theflat platform, with θ = 30 for various Ohnesorge numbers.

Fig. 4.13 shows reasonably good asymptotic agreement between the Tanner law and the nu-

merically computed spreading rate.

4.4.3 Droplet Impact Onto a Cylinder

In this test case, a planar water droplet in an air environment (ρw/ρa ≈ 800, µw/µa ≈ 50), with

a diameter of 1 cm, is placed above a cylinder of the same diameter. The droplet is initially

at rest, and is allowed to fall under the influence of gravity until impacting the cylinder. Two

different contact angles are compared, 30 and 150; results are illustrated in Fig. 4.14 and Fig.

4.15, respectively, demonstrating the effect wettability has on the impact dynamics.

Although it is not possible to create a planar droplet impact in reality, similar experiments

have been carried out for droplets onto spherical objects [57]. Droplet impacts are typically

characterized by two different phases: a fast, inertia driven phase followed by a slow phase

driven by capillary forces [58]. Comparison of the initial impact phase in Fig. 4.14 and Fig.

86

Figure 4.14: Planar droplet impact onto a hydrophilic (θ = 30) cylinder. Frames correspondto t = 0 s, t = 0.05 s, t = 0.07 s, t = 0.09 s and t = 0.2 s.

Figure 4.15: Planar droplet impact onto a hydrophobic (θ = 150) cylinder, at t = 0 s, t = 0.05s, t = 0.07 s, t = 0.1 s and t = 0.2 s.

4.15 at t = 0.05 reveals an initial spreading that is similar in both cases, as inertial effects

overwhelm the capillary forces. As the droplet spreading slows and wetting forces become more

dominant, the two cases diverge. The hydrophilic cylinder is covered completely in a thin liquid

film, while the liquid on the hydrophobic cylinder beads up before separating completely from

the cylinder.

4.4.4 Droplet Penetration into a Porous Media

As a final illustration of the efficacy of the method presented here, the 2D planar droplet

penetration into a porous media simulation studied by Liu and Ding [24] is presented. A

schematic of the setup for the problem is shown in Fig. 4.16. For this case, dimensionless

parameters are used. The droplet has a diameter of 1.5D; cylinders of diameter D are arranged

at (1.5, 0.9), (2.7, 0.9), (0.7, 2.1), (3.5, 2.1), and one cylinder of diameter 0.5D is placed at

(2.1, 3.05). The domain is discretized on a uniform mesh of dimensions 320× 400. The droplet

87

starts from rest, falls under the influence of gravity only. The dimensionless parameters are

defined in Table 4.1. The characteristic velocity u0 is defined to be the droplet speed just before

impacting the first cylinder.

λρ = ρ1ρ2

λµ = µ1µ2

Re = ρ2u0Dµ2

We =ρ2u20Dσ Fr = u0√

gD

0.001 0.025 62 1 0.44

Table 4.1: Non-dimensional parameters for droplet penetration into a porous media.

150

150 30

30 150

4.2D

5D

1.5D

Figure 4.16: Droplet penetration into porous media case.

As can be seen in Fig. 4.17, the droplet is split symmetrically by the first hydrophobic

cylinder. The symmetry is broken when the interfaces on either side make contact with the

top left hydrophilic cylinder and the top right hydrophobic cylinder. Eventually the two halves

separate completely, and a small satellite droplet is formed on the top hydrophobic cylinder,

88

Figure 4.17: Penetration of a planar droplet into multiple cylinders of varying wettability;snapshots at t∗ = 5, t∗ = 8, t∗ = 10, t∗ = 15, t∗ = 20 and t∗ = 30.

which is repulsed. Finally, the two halves come to rest, and only the hydrophilic cylinders

remain “wet”.

4.5 Conclusion

A novel algorithm for imposing contact angles on ghost-cell immersed boundaries was presented.

The algorithm combines sharp-interface, ghost-cell immersed boundaries [33] and the CELESTE

method for evaluating curvatures [32]. The ability of the algorithm to handle surfaces of differing

wettability was demonstrated. Thus, rather than imposing an angle at which the interface

intersects a boundary, the curvature itself is computed by incorporating the orientation of the

contact line, resulting in a momentum source that accounts for varying wettability.

Future work will focus on extending the proposed methods to moving boundaries in order to

study particle interactions at fluid-fluid interfaces, as well as an extension to 3D. In principle the

89

method presented should extend to 3D in a relatively straight-forward manner. One additional

challenge is that in 3D, there is not a single unique tangent vector along the IB surface, thus

determining the contact line normal along the IB surface requires information from the most

recent iteration of the VoF field. This interface vector must then be projected onto the IB

surface first in order to determine the contact line normal. Additionally, in 3D a hexagonal cell

has 26 neighbouring cells on its faces, edges and vertices, as opposed to only 8 neighbours in

2D. Thus, extending to 3D will require some numerical experimentation in order to determine

which Taylor series terms should be retained in the least-squares procedure.

90

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97

Chapter 5

A Moving Immersed Boundary

Method for Simulating Particle

Interactions at Fluid-Fluid Interfaces

With minor changes to fulfill formatting requirements and consistency with the symbology in the rest of thiswork, this chapter is largely the same as the manuscript submitted to the Journal of Computational Physics.

98

5.1 Abstract

A sharp interface immersed boundary method coupled to a volume-of-fluid method and suitable

for moving boundary problems is presented. A discrete-forcing approach is used, in which the

velocity reconstruction in the near-boundary region is performed using piece-wise continuous

bi-quadratic functions. A method is proposed for coupling the volume-of-fluid equation to

the immersed boundary method that allows for the boundary to move, as well as appropriate

boundary conditions for imposing contact angles. Numerical simulations of both static and

moving boundary problems are performed, which demonstrate the efficacy of the proposed

method. Validation is performed by comparing numerical results with experimental data and

theoretical models available in the literature.

5.2 Introduction

Two-phase flows involving the wetting/de-wetting of suspended particles are ubiquitous. Ex-

amples include the flotation recovery of coal, oil or minerals [1–3], surfactant-free emulsions

[4], drug delivery via nano- and micro-particles [5], removal of contaminants from soil [6], and

porous media flows [7]. In all of these flows, particles or geometric features of various shapes and

sizes interact with interfaces between different fluid phases. Understanding these interactions

involves the modelling of complex coupled non-linear interactions between the surface tension

forces at interfaces, solid-body motions of the particles, and hydrodynamic forces.

This work presents an Immersed Boundary Method (IBM) coupled to a moving contact

line model implemented in a 2D cartesian/axisymmetric Volume-Of-Fluid (VOF) framework,

to simulate flows involving moving particles and their interactions with fluid-fluid interfaces.

IBMs offer an advantage for problems involving the complex motion of boundaries with few or

no constraints, when a strategy involving re-meshing or dynamic meshes would be impractical.

In this work, we demonstrate a novel sharp-interface IBM for moving body simulations, with

specially developed contact line boundary conditions.

Much of the literature where IBMs are used to model complex geometries in multiphase

flow simulations has focused on high Weber (We) and capillary (Ca) number applications, in

99

which the effects of interfacial surface tension can be assumed negligible. For example, Zhang

et al. [8] coupled IBM and VOF models to study ocean engineering problems involving moving

structures. In low Ca/We regimes, however, surface tension, and by extension wetting/de-

wetting forces, may be the dominant driving forces in the flow, and special consideration must

be given to their modelling.

While many examples exist in the literature where contact lines are imposed on fixed mesh

boundaries [9–11], only a handful of authors have looked at imposing contact lines on Immersed

Boundaries (IBs), and fewer still have extended this to moving boundary problems. Deen et

al. [12] developed an IBM to study the interactions of rigid particles with bubbles/drops modeled

using the front-tracking method, and they later developed a slightly different method to study

particle impacts on thin liquid films [13]; Patel et al. [14] then extended this model to include

contact angle effects. More recently, Liu and Ding [15] developed a three-component diffuse-

interface IBM with contact angle effects, and later extended it to moving boundary problems

[16].

In the current work, we present an IBM method which can model moving solid boundaries

of varying hydrophobicity, utilizing a sharp-interface IBM which localizes the forcing of the

boundary as much as possible. In this method, a single set of momentum, continuity and

pressure equations is solved in both the fluid and solid domains, which has many advantages

for moving boundary problems, and also permits the use of common interface advection and

surface tension computation techniques.

The structure of this paper is as follows. In section 5.3, the equations governing the dynamics

of fluid-particle interactions are presented. In section 5.4, a numerical method is proposed to

approximately solve the governing equations. In section 5.5, simulation results are shown which

demonstrate the accuracy and applicability of the proposed numerical method. Finally, section

5.6 summarizes the paper and looks ahead to future research.

100

5.3 Governing Equations

5.3.1 Fluid Equations

The governing equations consist of the incompressible Navier-Stokes equations with additional

body forces representing surface tension and the presence of IBs. The momentum equation

takes the form,

∂ρu

∂t+∇ · ρuu =∇ · (τ − pI) + ρg + fσ + fib (5.1)

where ρ is the fluid density, u is the velocity field, p is the pressure, I is an identity tensor, g is

the gravitational acceleration, fσ is the surface tension force, fib is the body force imposed by

IBs, and τ is the viscous stress tensor which takes the form,

τ = µ(∇u +∇uT

)(5.2)

where µ is the dynamic viscosity. fσ can be further decomposed as,

fσ = −σκδn (5.3)

where σ is the surface tension coefficient, κ is the interface curvature, δ is the Dirac delta

function, and n is an outward pointing interface unit normal. The continuity, or conservation

of mass equation, takes the form,

∂ρ

∂t+∇ · ρu = 0 (5.4)

In the VOF method for incompressible flows consisting of two immiscible phases, ρ is ex-

pressed as a function of the volume fraction γ ∈ [0, 1],

ρ = ρ1 + γ(ρ2 − ρ1) (5.5)

where the subscripts 1 and 2 denote the first and second phases respectively. Substitution of

Eq. (5.5) into (5.4) yields the VOF advection equation,

101

∂γ

∂t+∇ · uγ = 0 (5.6)

Finally, µ can be computed by taking the harmonic average of the kinematic viscosity [17],

ρ

µ=ρ1

µ1+ γ

(ρ2

µ2− ρ1

µ1

)(5.7)

The Continuum Surface Force (CSF) method of Brackbill et al. [18] allows fσ to be conve-

niently expressed as a function of γ,

fσ = σκ∇γ (5.8)

without changing the value of fσ when integrated over a volume that encloses the interface.

Finally, in order to close the system of equations, the velocity field of two incompressible phases

is assumed to be divergence-free, yielding,

∇ · u = 0 (5.9)

which can be used in conjunction with Eqs. (5.1) and (5.6) to close the system of equations

through an appropriate numerical scheme, such as the projection method of Chorin [19].

5.3.2 Solid Equations

The motion of particles is governed by Newton’s second law. The linear acceleration can be

expressed as,

x =F

M(5.10)

where x is the position vector of a body’s centroid, F is the net force acting on the body, and

M is the mass of the body. The angular acceleration can similarly be expressed as,

θb =T

J(5.11)

where θb is the angular displacement of the body, T is the torque and J is the moment of

102

inertia. The evaluation of F acting on the body consists of hydrodynamic, gravitational and

capillary components, and can be expressed as,

F =

∮S

(τ − pI) · ns dS +Mg + Fσ (5.12)

where S denotes the closed IB surface, ns is an outward pointing surface unit normal, and Fσ

is the capillary force which can be expressed as,

Fσ = σlδt (5.13)

where t is the unit vector tangent to the interface that points away from the solid surface, l is

the length of the contact line, and the Dirac delta function δ takes the form,

δ =

1 x = xcl

0 otherwise

(5.14)

where xcl denotes the location of a contact line along the boundary surface. T can be computed

as

T =

∮S

(x− xc)× (τ − pI) · ns dS +

∫Ω

(x− xc)× ρsg dΩ +∑

(xcl − xc)× Fσ (5.15)

where x is the position vector, xc is the centroid of the body, ρs is the solid density, and Ω

defines the volumetric region occupied by the body.

5.4 Numerical Methodology

5.4.1 Discretization of the Fluid Equations

Eq. (5.1) is discretized and computed using a predictor/corrector scheme similar to that of Yang

and Stern [20], but with a modified bi-quadratic velocity reconstruction which will be discussed

in section 5.4.2. First, an initial predictor of Eq. (5.1) is constructed without the boundary

forcing term, fib, which takes the form,

103

ρn+1u∗c − ρnunc∆t

=− 1

2Ωc

∑f

[Fn + Fn−1

]+

1

2Ωc

∑f

[D∗ +Dn]

− ρn+1

[⟨∇pρ

⟩nf→c

+

⟨fσρ

⟩nf→c

+ g

] (5.16)

where Ωc is the volume of a cell, F and D denote the discretized fluxes for advection and

diffusion respectively, and the operator 〈·〉f→c denotes an appropriate face-to-cell interpolation

of the source term to the cell center [21]. D is computed semi-implicitly in order to allow larger

time-steps, although it is recognized that this procedure can reduce the temporal order of the

pressure [22]. In the second part of the fractional-step projection scheme, a corrector is applied

to Eq. (5.16) at the cell centers,

u∗∗c = u∗c + ∆t

(⟨∇pρ

⟩nf→c−⟨

fσρ

⟩nf→c

+fib

ρn+1

)(5.17)

where,

fib =ρn+1(uib − u∗c)

∆t(5.18)

and uib is the reconstructed velocity within the forcing region of the IB, and is zero everywhere

else. For collocated variable arrangements, the cell-centered velocities are interpolated to the

faces using a momentum interpolation [21],

u∗∗f = αu∗∗cl + (1− α)u∗∗cr +∆t

ρn+1f

[fσ]n+1f (5.19)

where the subscripts cl and cr denote the cells on either side of the face, and α denotes a linear

interpolation coefficient, which is computed from inverse distance weighting. In the next step,

a discrete Poisson equation is solved, which takes the form,

∇ · ∆t

ρn+1∇pn+1 =∇ · u∗∗ (5.20)

which then leads to the projection step [19] to enforce the divergence-free condition (Eq. (5.9))

104

by correcting the velocity field at the faces and cell centers,

un+1f = u∗∗f −

∆t

ρn+1f

∇pn+1f (5.21)

un+1c = u∗∗c −∆t

(⟨∇pρ

⟩n+1

f→c−⟨

fσρ

⟩n+1

f→c

)(5.22)

In the next section, a discrete, indirect forcing [23] approach to computing fib is presented.

5.4.2 Immersed Boundary Method

The IB method presented in this work is based on the direct-forcing method originally proposed

by Mohd-Yusof [24]; it uses a cell-based quadratic least-squares reconstruction of the forcing

term adjacent to the boundary, which is somewhat similar in concept to the face-based least-

squares method proposed by Martins et al. [25].

All cells which have at least one face neighbour with its cell center in the IB are flagged as

forcing cells. All node and face neighbour fluid and IB cells are added to the stencil. Addi-

tionally, the nearest point on the IB surface is added to the stencil (herein referred to as the

boundary point), as well as the boundary points of the other IB cells included in the stencil.

Fig. 5.1 depicts the reconstruction stencil.

The result of this procedure is a series of bi-quadratic functions which are piecewise contin-

uous. The forcing term, given by Eq. (5.18), depends on the value of its neighbours, and thus

the system must be solved implicitly. This system of equations takes the form,

∆t

ρn+1fib =

∑βi

(∆t

ρn+1fib + u∗

)i

+∑

βjub + u∗ (5.23)

where βi is the interpolation coefficient for the ith stencil cell and βj is the interpolation coeffi-

cient for the jth compatibility point on the boundary. ub must be interpolated from nearby cells

and boundary points. In the case of a fixed boundary velocity, the interpolation coefficients β

can be computed by forming a Vandermonde matrix of the form,

105

(xj , yj)

uib

(xi, yi)

Figure 5.1: Least-squares stencil ( ) and neighbouring least-squares stencils ( ) constructedby expanding a bi-quadratic polynomial to neighbouring fluid and forcing points. Dark graycells are solid cells, light gray cells are forcing cells.

β =

x2c

y2c

xcyc

xc

yc

1

T

x2i,1 y2

i,1 xi,1yi,1 xi,1 yi,1 1

......

......

......

x2i,Ni

y2i,Ni

xi,Niyi,Ni xi,Ni yi,Ni 1

x2j,1 y2

j,1 xj,1yj,1 xj,1 yj,1 1

......

......

......

x2j,Nj

y2j,Nj

xj,Njyj,Nj xj,Nj yj,Nj 1

−1

(5.24)

where (xc, yc) is the coordinate of the cell center, (xi, yi) are the coordinates of the face and

diagonal neighbours of the forcing cell, and (xj , yj) are the coordinates of the boundary com-

patibility points as depicted in Fig. 5.1.

The above procedure is valid for fixed boundary conditions. Boundary conditions for the

pressure field are not directly imposed on the IB; rather the projection step is performed over

the entire domain, including within the solids. It is well documented in the literature that failure

to uphold geometric conservation of the fluid domain by solving it separately from the solid

domain in moving boundary simulations can lead to spurious oscillations in the pressure field

106

[26, 27]. The most straightforward way to avoid violating geometric conservation is to simply

solve Eqs. (5.16) and (5.20) over a unified domain. This procedure is valid since solid-body

motion is divergence-free.

5.4.3 Immersed Boundary/Volume-of-Fluid Contact Line Extension

Imperative to the simulation of fluid-fluid interfaces near the surface of an IB is the treatment

of the contact line. The presence of an IB will result in cells that are partially embedded within

the boundary, and may not completely fill up with fluid as the contact line passes through them.

This presents a problem, as it can result in artificial interfaces between the phases along the

IB surface, which will result in an artificial computation of interface forces, as well as incorrect

hydrodynamic forces acting on the body. Additionally, the CSF method [18] requires the

convolution of the nearly discontinuous γ field in order to reasonably approximate the interface

normal n. Liu and Ding [15] addressed this issue by introducing a characteristic Moving Contact

Line (MCL) which would extend the interface into the boundary by interpolating the liquid

volume fractions along the contact line, using a one-dimensional quadratic function. Patel et

al. [14] also recognized the need to extend the volume fraction field into the IB, but did so using

an advection equation with a prescribed pseudo-velocity for γ within the solid region itself. In

this work, we adopt a similar characteristic approach to that of Liu and Ding [15], where we

extend the volume fractions along the contact line into the IB by a distance equal to the radius

of the convolution function used to smooth the γ field before computing the interface normals

[18]. In this procedure, two stencils M1 and M2, which are symmetric about the outward

pointing solid unit normal ns, are constructed, as depicted in Fig. 5.2.

An A∗ traversal is then used to find a point where the contact line intersects a horizontal,

vertical or diagonal line segment which connects the centers of two adjacent cells. This results

in the volume fraction fields along IBs depicted in Fig. 5.3.

In this work, the advection of γ is carried out using the Compressive Interface Capturing

Scheme for Arbitrary Meshes (CICSAM) algebraic interface advection scheme of Ubbink and

Issa [28]. In this scheme, a variation of second-order Crank-Nicolson time advancement is used,

and the discretized form of Eq. (5.6) becomes,

107

θθ

ns

c1 c2~p

M1 M2

Figure 5.2: Contact line interpolation stencils for extending the γ field into the boundary.

Fluid Region

Solid Region

(a)

Fluid Region

Solid Region

(b)

Figure 5.3: Result of the contact line extension near an IB for (a) θ = 45 and (b) θ = 135.The extension is performed for all cells between the extension boundary ( ) and IB surface( ).

γn+1 − γn

∆t+

1

2Ωc

∑Fnf

[γn+1f + γnf

]= 0 (5.25)

108

where the details regarding the interpolation of γf are given in [28]. First, an explicit Euler

predictor of the form,

γ∗ − γn

∆t+

1

Ωc

∑Fnf γ

nf = 0 (5.26)

is solved, and a mass source of the form,

γ =γcl − γ∗

∆t(5.27)

is then computed, where γcl is determined from the contact line extension procedure described

above. Finally, a modified version of Eq. (5.25) is solved, which has the form,

γn+1 − γn

∆t+

1

2Ωc

∑Fnf

[γn+1f + γnf

]= −γ (5.28)

5.4.4 Immersed Boundary Wetting Force

The procedure for computing the modified curvature, through which the wetting force is im-

posed, differs slightly from our previous work in [29]. In the current work, curvature is computed

using a modified CELESTE scheme [30], the details of which are outlined in [29]. However, the

curvature stencil is not modified near the IB. Instead, interface normals are prescribed for the

cells in the contact line extension region (Fig. 5.3) directly from the stencil used to extend γ

into the solid.

5.4.5 Computation of Solid Body Forces

The net forces acting on an IB are described by Eq. (5.12). In this section, details pertaining

to the computation of the hydrodynamic and capillary forces will be presented.

Hydrodynamic Forces

The momentum equation, Eq. (5.1), integrated over a volume Ωs that encloses a solid body,

takes the form,

109

∫Ωs

Dρu

DtdΩs =

∮∂Ωs

(τ − pI) · n d∂Ωs +

∫Ωs

(fσ + fib + ρg) dΩs (5.29)

The hydrodynamic force acting on the region Ωs can be obtained by re-arranging Eq. (5.29),

∮∂Ωs

(τ − pI) · n d∂Ωs =

∫Ωs

(Dρu

Dt− fσ − fib − ρg

)dΩs (5.30)

which when discretized takes the form,

Fh ≈∑Nib

(ρn+1un+1 − ρnun

∆t+

1

2

[Fn + Fn−1

]− fib − fσ − ρn+1g

)Ωc (5.31)

where Nib denotes the number of solid and forcing cells associated with a particular IB. In

general, the volume integral on the right-hand-side in Eq. (5.30) is easier to compute numerically

than the surface integral on the left-hand-side. This is especially true when performing parallel

computations.

Capillary Forces

Finally, the capillary forces acting tangentially to the contact line must be accounted for. Liu

and Ding [16] proposed a hybrid method that switches between a sharp and diffuse model based

on how close the interface profile is to equilibrium. In this work the sharp representation of

the interface means that the capillary force can be computed reliably from a sharp model for

all but very large We. Once the orientation of the interface is known, the appropriate stencil

from the candidates M1 and M2 (Fig. 5.2) can be chosen. These stencils can also be used to

project γ onto discrete points along the IB surface. These points can then be used to perform an

interpolation as depicted in Fig. 5.4. In order to locate the intersection point, an interpolation

is performed along an arc in order to find the relative angular displacement,

∆φ = φB +

(0.5− γBγA − γB

− 1

)φA (5.32)

where φA and φB are the angular locations of the intersections of xA and xB respectively. The

point of intersection for a circular surface is then found simply by rotating the vector xA−xC ,

where xC is the location of the centroid of the IB, by ∆φ,

110

xA

xBxp

tcl

∆φ

Figure 5.4: Interpolation stencil for the sharp interface capillary force calculation.

xp = xc +

cos ∆φ − sin ∆φ

sin ∆φ cos ∆φ

(xA − xC) (5.33)

and the direction of the capillary force can then be computed as,

tcl =

cos ∆φ − sin ∆φ

sin ∆φ cos ∆φ

tA (5.34)

After the location and direction of the capillary force are identified, the capillary force can

be computed using Young’s equation,

~Fσ =∑Ncl

σltcl (5.35)

where Ncl denotes the number of contact lines, and l is contact line length.

5.4.6 Solution Procedure

The solution procedure proceeds as follows:

111

1. Solve the VOF equations, Eqs. (5.26) and (5.28), and update γ to the n+ 1 time level.

2. Compute all properties at the n+ 1 time level which are functions of γ, which include ρ,

µ and fσ.

3. Solve the momentum equations, Eq. (5.16)-(5.19), including the IB forcing equation, Eq.

(5.23).

4. Solve the pressure Poisson equation, Eq. (5.20).

5. Perform the projection step by correcting the velocities on faces and cell centers, using

Eqs. (5.21) and (5.22).

6. Compute the forces acting on the IB from Eqs. (5.31) and (5.35), as well as the known

weight of the IB.

7. Update the IB translational and angular positions.

5.5 Results

To evaluate the proposed methodology, several test cases are considered. In the first case, the

hydrodynamic forces are validated by simulating an inline oscillating cylinder with prescribed

motion. The second case is of an axisymmetric spherical droplet situated on a spherical IB, and

is a demonstration of the algorithm’s ability to impose the desired equilibrium contact angle.

The third case looks at the sinking of a horizontal cylinder, and shows that the balance of

hydrodynamic and capillary forces are reasonably predicted. A similar validation is performed

for spherical objects using the axisymmetric version of the algorithm. Both the sinking cylinder

and sphere cases include fluid-fluid interfaces and fluid-particle interactions and present a chal-

lenge to a non-boundary conforming methodology. The final case illustrates multiple particles:

the formation and destabilization of a cylindrical particle raft on a fluid-fluid interface.

5.5.1 In-Line Oscillation of a Cylinder

We begin with a validation of the hydrodynamic force calculation by prescribing motion to an

in-line oscillating cylinder. This test case has been studied extensively in previous work [20,

112

31, 32]. A cylinder is placed at the center of a domain of 40D × 24D, where D is the cylinder

diameter, and oscillates in the x direction according to,

x(t) = x0 −A sin (2πft) (5.36)

where x0 is the initial x coordinate position, A is the amplitude of oscillation and f is the

oscillation frequency. The problem is parameterized with respect to the Keulegan-Carpenter

number KC,

KC =|U|fD

(5.37)

where |U| is the characteristic velocity, defined as the maximum cylinder velocity. Here we

consider KC = 5 and Re = 100. The time-step is fixed at ∆t = 5 × 10−3 for three different

grid resolutions, so that the spatial convergence can be studied. It is well known that moving

sharp-interface IBMs can suffer from so-called spurious force oscillations [26, 32]. In Fig. 5.5,

the drag coefficient CD = Fx/(ρ|U|D) for three different grid resolutions is compared to the

experimental data of [31]. The spurious force oscillations, which are strongest during maximum

acceleration, decay rapidly as the grid resolution is increased. In Fig. 5.6, the pressure and

vorticity contours for the oscillating cylinder are shown.

t/T

CD

0 0.2 0.4 0.6 0.8 1­2

­1.5

­1

­0.5

0

0.5

1

1.5

2

Figure 5.5: Time history of the drag coefficient for ∆h = 0.04D ( ), ∆h = 0.02D ( ),∆h = 0.01D ( ), compared against the experimental data () in [31].

113

Good agreement between the IBM and the experimental data is obtained. Periodic, sym-

metric vortices form as the cylinder oscillates back and forth, and dissipate as the cylinder slows

before reversing direction. Relatively strong vorticity is present upon reversing direction, since

the cylinder interacts with its own wake.

5.5.2 Droplet on a Static Sphere

A spherical droplet is placed on top of a solid sphere, initially perturbed from the equilibrium

position, and is allowed to evolve until a stable configuration is obtained. The exact equilibrium

configuration in the absence of gravity can be determined from geometrical arguments. The

volume of the droplet is given by,

Vd =4

3πr3

d −π(rs + rd − l)2(l2 + 2rd − 3r2

d + 2lrs + 6rdrs − 3r2s)

12l(5.38)

where Vd is the known droplet volume, rd is the radius of curvature of the droplet, rs is the

radius of the solid sphere and l is the spacing between the center of curvature of the dropet and

the center of the solid sphere. The contact angle θ is then,

θ = acos

(r2s + r2

d − l2

2rsrd

)(5.39)

In order to determine the droplet equilibrium radius rd and the spacing l, the roots of Eq.

(5.38) and Eq. (5.39) are found by applying an iterative Newton-Krylov procedure. In Fig. 5.7,

the final configurations of the numerically computed equilibrium states are compared against

the exact solutions described above. Initially, the droplets are positioned such that the interface

intersects the surface of the solid sphere at θ = 90, and the solution is allowed to proceed until

the norm of the velocity solution vector is sufficiently small.

The numerically determined internal droplet pressures and the droplet radii are shown in

Fig. 5.8a and Fig. 5.8b respectively.

The numerical model is able to successfully predict the equilibrium configurations of spher-

ical droplets over a wide range of θ. It is well-known that when imposing no-slip boundary

conditions on u, a stress singularity at the contact line is the result [33]. Ideally, in order to

obtain mesh independence of the droplet spreading rate, the numerical model should include

114

a dynamic contact angle, such as the one proposed by Afkhami et al. [34]. We leave this to a

future study.

5.5.3 Sinking Cylinder

Here, the prediction of a freely moving IB at a fluid-fluid interface is demonstrated, and com-

pared to the experimental results and theoretical model of Vella et al. [35]. In their work, a

hydrophobic glass cylinder was placed at a fluid-fluid interface, and then allowed to sink, and

the angular position of the contact line β and the non-dimensional height H0 were measured.

A schematic of the numerical set-up, which approximates the experimental set-up in [35], is

given in Fig. 5.9.

In [35], an analytical model was derived which approximates the dynamics of the sinking

cylinder using potential flow theory, with an additional force to represent the capillarity. The

analytical model takes the form,

DπR2H0 = Fp − πR2D + 2 sinφ (5.40)

where D is the density ratio between the solid cylinder and the liquid, R is the non-dimensional

cylinder radius with respect to the capillary length scale lc =√σ/(ρ2|g|), H0 is the non-

dimensional distance of the cylinder center below the free surface, and φ is the angle formed

between the free surface and the horizontal plane at the contact line, and is related to the

contact angle through the geometric relationship φ = θ+β−π, where β is the angular position

of the contact line, and Fp is the vertical pressure force given by,

Fp = −2H0R sinβ +R2(β + cosβ sinβ)+

R2(cosβ sinβ − β)H0 +8

3H2

0 sin3 β

(5.41)

The characteristic time for this case is defined as tc =(σ/(ρ2|g|3

))1/4. In order to close

Eq. (5.40), a line is fit to the experimental data in [35] for the angular contact line position β.

For the numerical solution, a horizontal cylinder was placed at the interface, with an initial

115

velocity estimated from the experimental data, and allowed to sink. The contact angle imposed

on the surface of the cylinder is fixed at 111, which was also used in [35] for the analytical

model. Fig. 5.10 is a qualitative comparison of the numerical results and the experimental

images of Vella et al. [35].

The numerically computed, experimentally observed, and analytically predicted positions

of the cylinder centroid with respect to the undisturbed free surface, and the angular position

of the contact line are compared in Fig. 5.11.

Good agreement is demonstrated between the analytical and numerical simulations, al-

though both deviate somewhat from the experimentally observed positions, which is most likely

due to the assumption of a constant contact angle. Since the cylinder is denser than the liq-

uid, it slowly sinks. For reference, in Fig. 5.11, the analytical solution with the capillary force

omitted is included to show the impeding effect of the surface tension on the rate at which the

cylinder sinks. As seen in Fig. 5.11b, at T u 4.5, the cavity forming behind the cylinder closes,

and the capillary force which is acting to slow the decent of the cylinder rapidly goes to zero,

and the cylinder enters free-fall.

5.5.4 Impact of a Superhydrophobic Sphere

Lee and Kim [36] studied the impacts of superhydrophobic spheres (θ ≈ 150). Depending on

We and Bond number (Bo), a sphere impacting a liquid-liquid interface will either sink, bounce

or oscillate due to the presence of capillary forces. Scaling laws were used to determine the

relationship between We, Bo and the solid-to-liquid density ratio D = ρsρl

. It was reported that

boundaries between sinking and bouncing, as well as between bouncing and damped oscillation

could be represented asymptotically by,

WeBo3/2 ∝ D−2 (5.42)

Axisymmetric numerical simulations were carried out for two of the experiments conducted

in [36]. The numerical set-up is depicted in Fig. 5.12. The sphere radius and impact velocity are

taken as the characteristic length and velocity respectively. Fig. 5.13 presents results for two

cases, a sinking sphere and bouncing sphere. For the sinking sphere, We = 12.7, Bo = 0.407,

116

and D = 1.99, while for the bouncing sphere, We = 20.9, Bo = 0.496 and D = 1.32. In both

cases, the spheres have a fixed contact angle of 154, representing the superhydrophobic coating

that was applied in the experiments.

Good agreement with the experimental results is demonstrated for both cases, although the

bouncing sphere appears to be slightly affected by waves reflecting off of the r = 0 and r = R

boundaries. Nonetheless, the proposed method is able to predict the transition from sinking to

bouncing with the relevant non-dimensional parameters. The sinking and bouncing behaviours

are depicted in Figs. 5.14 and 5.15 respectively.

In both cases, the initial deceleration after impact is dominated by strong hydrodynamic

forces. As the sphere slows, the force balance becomes increasingly dominated by capillary and

buoyancy forces. In the case of the sinking sphere, the cavity behind the sphere collapses, and

the sphere sinks. In the case of the bouncing sphere, the relatively higher capillary forces cause

the sphere to begin accelerating upwards, and the sphere is ejected from the surface. According

to [36], at even lower We or Bo, the sphere would simply become trapped at the interface and

proceed to oscillate back and forth.

5.5.5 Formation of Cylindrical Granular Rafts

Dense particles at interfaces can self-assemble into rafts at oil-water interfaces. Although heavier

than either phase, the vertical component of the capillary force can keep them suspended at

the interface. With the addition of more particles, the horizontal component of the capillary

force may pull the particles together, forming a so-called ‘raft’. As the particles move closer

together, the curvature of the interface between two particles may increase, which results in

an increased capillary pressure. This may cause an interface to retract from between two

particles. This can de-stabilize the vertical capillary force, resulting in the raft sinking. This

effect was studied experimentally by Abkarian et al. [37]. Although this phenomenon is three-

dimensional in nature, it can be demonstrated qualitatively using the present framework. Here,

three cylindrical particles are suspended above an oil-water interface and allowed to sink until

they reach the interface. The capillary forces are strong enough that the particles become

temporarily trapped, while the horizontal component of the capillary forces assembles them.

The assembly of the raft results in a loss of vertical capillary force, and then the raft sinks,

117

entraining the oil phase with it. The numerical set-up is depicted in Fig. 5.16. Snapshots of

the simulation are depicted in Fig. 5.17.

A simple collision model, described by Glowinski et al. [38], which uses a repulsive force to

prevent particle-particle and particle-wall interpenetration, was implemented. Additionally, as

particles come very close together, the reconstruction stencil depicted in Fig. 5.1 defined by Eq.

(5.24) is modified simply by adding additional compatibility points between the two boundaries.

The present method is able to predict both the assembly of the raft due to capillary forces, as

well as its eventual sinking due to the destabilization of the vertical component of the capillary

force.

5.6 Conclusions

A sharp-interface discrete immersed boundary method with indirect forcing, coupled to a VOF

multiphase solver with contact line dynamics has been presented for the simulation of objects

at fluid-fluid interfaces. This model uses polynomial velocity reconstructions in cells near the

immersed boundary to compute a forcing term, similar to previous sharp-interface approaches

in the literature. The computation of the wetting force near the boundary is similar to the

approach in [29], where modified interface normals near the boundary are used in the curvature

computation, and the VOF field is extended directly into the boundary.

Select model problems demonstrate that the proposed method can accurately simulate con-

tact line dynamics on both static and moving immersed boundaries. The ability to perform FSI

simulations with the influence of capillary forces is also demonstrated. Future work will inves-

tigate mesh independent contact line models, as well as dynamic contact angles. In principle,

extending the proposed framework to three dimensions should be relatively straightforward, as

there are many examples of similar three dimensional IBs found in the literature.

118

t/T = 0

t/T = 0.25

t/T = 0.5

t/T = 0.75

Figure 5.6: Pressure contours (left) and vorticity magnitude countours (right) for the inlineoscillating cylinder, with KC = 5 and Re = 100.

119

(a) (b) (c)

Figure 5.7: Final configurations of the spherical droplets situated on solid spheres, with (a)θ = 45, (b) θ = 90 and (c) θ = 135, and compared against the exact solution ( ).

θ

p*

50 100 1500.75

0.8

0.85

0.9

0.95

1

(a)

θ

rd

50 100

0.45

0.5

0.55

(b)

Figure 5.8: (a) Internal droplet pressure compared to the exact solution, and (b) droplet radiicompared to the exact solution.

.

120

12D

4D

1Dρgρl

= 0.001µgµl

= 0.02

Figure 5.9: Numerical setup for the horizontal sinking cylinder.

(a) T = 0.88 (b) T = 1.76

(c) T = 2.64 (d) T = 4.48

Figure 5.10: Snapshots comparing the numerically simulated and experimentally determinedpositions of a sinking horizontal cylinder.

121

T

H0

0 1 2 3 4 5­2.5

­2

­1.5

­1

­0.5

0

(a)

T

dH

0/dT

0 2 4 6

­0.6

­0.4

­0.2

Free surface detachment

(b)

Figure 5.11: Comparison of numerical results ( ) against the analytical solution with capillaryforces ( ), analytical solution without capillary forces ( ) and experimental data () in [35].

12R

8R

8R

ρgρl

= 0.001

µgµl

= 0.02

Figure 5.12: Numerical set-up for the axisymmetric superhydrophobic sphere impact.

122

T

H0

0 2 4

­1.5

­1

­0.5

0

0.5

(a)

T

H0

0 5 10

­1

­0.5

0

0.5

(b)

Figure 5.13: Numerical ( ) and experimental () non-dimensional heights for (a) the sinkingsphere and (b) the bouncing sphere.

(a) T = 1.76 (b) T = 3.53 (c) T = 5.29

(d) T = 7.05 (e) T = 8.81 (f) T = 10.58

Figure 5.14: Snapshots of the sinking sphere simulation.

123

(a) T = 1.63 (b) T = 3.25 (c) T = 4.88

(d) T = 6.50 (e) T = 8.13 (f) T = 9.75

Figure 5.15: Snapshots of the bouncing sphere simulation.

10D

5D

3D

2D 3D

1.5D

Figure 5.16: Set-up for the cylindrical granular raft simulation.

124

(a) T = 1.44 (b) T = 2.99

(c) T = 5.20 (d) T = 13.29

(e) T = 22.14 (f) T = 44.29

Figure 5.17: Snapshots of the cylinder raft formation.

125

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130

Chapter 6

Pore-scale direct numerical

simulation of Haines jumps in a

porous medium model

With minor changes to fulfill formatting requirements and consistency with the symbology in the rest of thiswork, this chapter is largely the same as the manuscript submitted to the International Journal of MultiphaseFlows.

131

6.1 Abstract

Direct numerical simulations are presented for a porous media model consisting of two immis-

cible fluids, an invading and defending phase, in a two-dimensional micro-geometry filled with

randomly sized and randomly distributed cylinders. First, interface instability and penetration

modes are studied when varying the wetting features of a single pore in the porous medium. It

is found that the displacement patterns not only change with the capillary number, as previ-

ously observed, but also are a function of the contact angle, even for a viscosity ratio of unity.

This is an important conclusion suggesting that capillary number and viscosity ratio alone

cannot completely describe the pore-scale displacement. Second, rapid pore-scale displacement

is considered, where the displacements are accompanied by sudden interface jumps from one

site to another, known as Haines jumps. The characteristic time and length scales of a Haines

jump are examined to better understand the transient dynamics of the jump. We then focus

on analyzing the Haines jump in a simple pore configuration where cylinders of equal size are

placed at the vertices of equilateral triangles. We use this geometry to provide more insight

into the effect of the contact angle at which the Haines jump is predicted.

6.2 Introduction

The flow of immiscible fluids in porous media has applications in subsurface water flows, tar

sands oil production, and CO2 sequestration, to name a few [1]. At the micro-scale, where the

capillary forces are dominant, the wetting effects can play a significant role in the displacement

dynamics; depending on wetting properties, the invading phase can completely drain the other

phase, or some of the displaced phase can become trapped by the invading phase [2–4]. The

trapping can be important as it affects the drainage efficiency. The influence of surface tension

and viscous effects on the drainage displacement patterns have recently been studied in the

literature, see e.g. [5, 6]. In [6], for instance, the authors investigate the effects of combined

viscous and capillary forces and the viscosity ratio and identify three main patterns: i) viscous

fingering, ii) capillary fingering, and iii) stable displacement. They show that the pattern

depends on the ratio of the invading to defending fluid viscosity and the capillary number, Ca,

132

which represents the relative magnitude of the viscous force to the surface tension force. They

show that viscous fingering occurs at high Ca and when the viscosity ratio is small, capillary

fingering occurs when Ca is sufficiently low, and stable displacement occurs when both Ca and

the viscosity ratio are high.

The study of the influence of the contact angle on different displacement patterns is impor-

tant, as it is also one of the key factors in defining the flow behavior, and therefore the resulting

fluid distribution within a porous media [4]. In the pioneering work of Cieplak and Robbins [7,

8], a two-dimensional quasistatic numerical model was developed to describe wetting conditions.

However, the approach in [7, 8] simplifies the interface representation as circular arcs connect-

ing the solid geometries, which in their work are two-dimensional array of disks with random

radii. Recent particle-based simulations account for wall wettability in two-dimensional [9] and

quasi-two-dimensional [10] models. However, particle-based simulations present difficulties in

enforcing the incompressibility in very viscous flows. Nonetheless, all the studies above con-

sistently show the transition from stable front displacement at low contact angles to capillary

fingering at high contact angles.

In contrast to approaches discussed above, direct numerical simulation techniques based on

solving the full Navier-Stokes equations provide robust and high accuracy solutions. However,

a great challenge in pore-scale direct numerical simulations is the inclusion of wetting effects

into the numerical model through the contact angle boundary condition. There exist direct

numerical studies on pore-scale modeling of wettability effects, such as the phase-field simulation

in [11] and the lattice Boltzmann simulation in [12]. Here we present the results of a direct

numerical method that is based on an interface capturing approach. While the majority of

previous (Eulerian) interface capturing methods include wetting on body-fitted meshes (see e.g.

[13, 14] and for a review [15]), less progress has been made to develop numerical methods that

handle arbitrary geometries. To that end, one attractive approach is the Immersed Boundary

Method (IBM) [16, 17], because of its ability to complex geometries, and its potential for

modeling systems of solid bodies with arbitrary relative motion [18]. Here we use an accurate

and robust numerical model developed in [19] and later extended in [18] for combining wetting

dynamics, in a Volume-of-Fluid (VOF) framework, and the IBM.

The result we present here is first of its kind in that it uses a combined VOF/IBM pore-

133

scale direct numerical simulations to quantify the effects of the contact angle and Ca on the

displacement phenomena in a porous media model. We examine how the contact angle controls

displacement patterns, and show the crossover from a stable propagating front to a fractal

pattern, namely a fingering instability. Our work extends on the literature by identifying

the displacement regimes as functions of independently varying Ca and θ. We illustrate the

interface patterns as we change the contact angle, and analyze the instabilities in terms of the

growth mechanisms and the rapid movement of interfaces between pores. Based on our results,

we describe three main patterns and the crossover from fingering to stable flow as a function of

Ca and the contact angle, at a viscosity ratio of 1. We show that decreasing the contact angle

results in a change of the flow behavior from strong fingering to stabilized front propagation,

and that the contact angle effect diminishes as Ca increases. The threshold for this transition

is related to the contact angle amongst other parameters. For wetting invading fluids, the front

propagates rather smoothly, draining all of the displaced phase, while a non-wetting invading

fluid can lead to residual fluid. Our simulations reveal how post-fingering production flux is

different above a critical contact angle from the one below the critical contact angle where the

front is stable, for sufficiently small Ca, and that the production flux is similar when Ca is

sufficiently large, irrespective of the contact angle. Our study therefore allows the quantification

of the flow mechanisms when including the wetting effects.

We also pay particular attention to sharp interfacial jumps during the displacement, known

as Haines jumps [7, 20, 21]. This phenomena can lead to a highly discontinuous rate at which

the front progresses, identified as bursting, when it would otherwise progress slowly and con-

tinuously. Here we numerically investigate a two-dimensional invading fluid displacing another

fluid in an arrangement of circular disks. We show the influence of the Haines jump on the

propagation of the front, which is accompanied by a sharp pressure oscillation. We further

investigate the effects of the interfacial tension and the wettability on the jump transient dy-

namics. Finally, we study the frequency of Haines jumps as a function of the contact angle and

Ca. This information can be useful to predict the residual saturation of the displaced phase.

134

Symmetry

Outlet

L

U

Figure 6.1: Domain set-up for the Haines jump simulations.

6.3 Computational Setup

In this work, the flow equations are solved numerically and the pore geometry is modeled

using an immersed boundary method [19] in place of body-fitted meshes. The incompressible

Navier-Stokes equations for a two-component fluid system are solved, where the interface is

tracked using a Volume-of-Fluid (VOF) method. A contact line boundary condition is imposed

on surfaces of the immersed boundaries. This approach has many advantages; in particular

it greatly simplifies the inclusion of complex domain geometries, as the need to produce a

body-fitted mesh is eliminated.

The Navier-Stokes equations for a two component fluid include the momentum equation,

∂ρu

∂t+∇ · ρuu = −∇p+∇ · µ

(∇u +∇uT

)+ fσ (6.1)

and the mass conservation equation,

∂ρ

∂t+∇ · ρu = 0 (6.2)

where ρ is the fluid density, µ is the fluid viscosity, u is the fluid velocity, p is the pressure, and

135

fσ is a surface tension body force.

For an incompressible flow consisting of two immiscible phases, the density is expressed

as ρ(γ) = ρ1 + γ(ρ2 − ρ1), while the viscosity is computed from harmonic averaging, ρ(γ)µ(γ) =

ρ1µ1

+γ(ρ2µ2− ρ1

µ1

), where γ is the fluid volume fraction governed by the VOF advection equation,

∂γ

∂t+∇ · uγ = 0 (6.3)

which is solved using a suitable numerical scheme that limits numerical diffusion in order to

maintain a sharp interface between the two fluids. In this work, the Compressive Interface

Capturing Scheme for Arbitrary Meshes (CICSAM) scheme of Ubbink and Issa [22] is used

to solve the discretized form of Eq. (6.3). The computation of fσ is carried out using the

Continuum Surface Force (CSF) method of Brackbill et al. [23], which models surface tension

as

fσ = σκ∇γ (6.4)

where σ is the constant surface tension coefficient, and κ is the interface curvature. In order

to close the system of equations, the divergence-free constraint is imposed on the velocity field,

which has the form

∇ · u = 0 (6.5)

Eqs. (6.1) and (6.5) are solved together using a fractional-step projection method [24],

based on the balanced-force algorithm of Francois et al. [25]. The details of the discretization

are omitted here, and can be found in [19]. For the simulations reported in this work, Eq. (6.1)

is non-dimensionalized as follows,

Re

(∂λρ ~u∗

∂t∗+∇∗ · λρu∗u∗

)= −∇∗p∗ +∇∗ · λµ

(∇∗u∗ +∇∗u∗T

)+

1

Caκ∗∇∗γ (6.6)

where the superscript ∗ denotes a non-dimensional quantity, λρ is the density ratio, λµ is the

viscosity ratio, and the Reynolds number Re and Capillary number Ca are defined as follows,

136

Re =ρ|U|Dµ

Ca =µ|U|σ

where |U| is the characteristic velocity, taken here to be the bulk flow velocity, and D is the

mean cylinder diameter, which is sampled from a normal distribution. In this work, density and

viscosity ratios are set to unity. The characteristic velocity, |U|, corresponds to Q/A, where Q

is the flow rate through the domain and A is the cross-sectional area of the domain.

6.4 Results and Discussions

The results are organized as follows. In Section 6.4.1, the primary mechanism of pore-filling,

the Haines jump, as well as its two modes, are investigated. In section 6.4.2, the different

displacement regimes are characterized as functions of Ca and θ. In Section 6.4.3, the effects of

the contact angle and capillary number on the shape of the propagation front of the invading

phase are examined.

6.4.1 Haines Jump Modes

A simplified setup is used in order to isolate the primary mechanisms of pore-filling. Three

different Ca are compared, and the results are illustrated in Figs. 6.2 to 6.4. Singh et al. [6]

identified two different geometry-dependent Haines jump modes for cylinders and spheres. In

2D, these modes effectively correspond to the ‘touching’ and ‘coalescence’ modes described in [7,

8]. In order to qualitatively demonstrate both modes, a simplified simulation domain, depicted

in Fig. 6.1 was constructed. As the interfaces corresponding to θ = 30, θ = 90, and θ = 150

pass between the two left-most cylinders, they begin to ‘bulge’ until they make contact with

the center cylinder (Fig. 6.2c). This formation of a new contact line on the center cylinder

initiates a rapid reconfiguration of the interface due to unbalanced surface tension forces. This

represents the first mode of the Haines jump. As Ca is decreased, and as the interfaces approach

the throats formed by the right-most cylinders, they merge with a secondary interface passing

around a cylinder (Figs. 6.3d and 6.4d). This causes a rapid release of the capillary pressure

137

T* = 0.2

(a)

T* = 0.4

(b)

T* = 0.58

Haines jump(touching)

(c)

T* = 0.68

(d)

T* = 0.75

(e)

T* = 0.9

(f)

Figure 6.2: Simulation results displaying the Haines jump mechanism for θ = 30 ( ), θ = 90

( ), and θ = 150 ( ) for Ca = 1× 10−3.

gradient acting against the flow, which causes the invading fluid to rapidly fill the pore and

move through the throat. This corresponds to the second mode of the Haines jump.

Both modes of Haines jump will only occur at certain values of Ca. As seen in Fig. 6.4, the

viscous boundary layer prevents a contact angle from forming on the middle cylinder before

the contact lines traversing the surface of the left-most cylinders merge. Conversely, at low Ca

(Fig. 6.2), the driving pressure may be too small to overcome the capillary forces at the throat,

and the interface stalls, effectively blocking flow through the throat.

The effect of the geometry, injection rate and capillary number on the time scale of the

Haines jumps were studied in detail by Armstrong et al. [26]. Whether or not a Haines jump

will occur can be predicted from simple geometrical arguments. If the media itself is very

hydrophilic to the invading fluid, and the media is not densely packed, then it is possible

that neither mode of Haines jump will occur. For the first mode, the contact lines on the

138

T* = 0.2

(a)

T* = 0.4

(b)

T* = 0.6

(c)

T* = 0.68

Haines jump(coalescence)

(d)

T* = 0.73

(e)

T* = 0.9

(f)

Figure 6.3: Simulation results displaying the Haines jump mechanism for θ = 30 ( ), θ = 90

( ), and θ = 150 ( ) for Ca = 2.5× 10−3.

adjacent sides of an obstacle will simply merge, and the protruding of the interface from the

throat necessary for the second mode to occur will not be observed. In this scenario, complete

displacement of the defending phase will be observed. When a Haines jump occurs, a pathway

forms for the fluid to travel, in which no resistance from capillary pressure is present. As a

result, pockets of defending fluid may remain in place, since the invading fluid will simply flow

around them through sites in which Haines jumps have occurred.

At higher Ca (Figs. 6.3 and 6.4), a liquid film may become trapped along the solid boundary,

which collapses into a droplet upon the formation of contact lines for the hydrophilic case.

Decreasing Ca results in a more stable contact line, with a θ that corresponds more closely to

the prescribed equilibrium value.

139

T* = 0.2

(a)

T* = 0.4

(b)

T* = 0.6

(c)

T* = 0.67

Haines jump(coalescence)

(d)

T* = 0.8

(e)

T* = 0.9

(f)

Figure 6.4: Simulation results displaying the Haines jump mechanism for θ = 30 ( ), θ = 90

( ), and θ = 150 ( ) for Ca = 5× 10−3.

6.4.2 Displacement Regimes

The primary motivation behind this work is to investigate the effect that differing cylinder

hydrophobicity has on the efficiency of the invasion of a fluid-fluid interface into a porous

media. In order to investigate this, a random array of randomly-sized cylinders was generated,

with different prescribed contact angles ranging from θ = 30 to θ = 150. The secondary phase

is injected into the cylinder array at a constant rate, at Re = 5, where the characteristic length

is defined as the mean cylinder diameter. The displacement regimes include viscous fingering

(high Ca), capillary fingering (low Ca, θ > 90) and stable displacement (low Ca, θ < 90).

The existence of these regimes has been discussed by previous authors [5, 10, 21, 27–29] for

different independent parameters. Injection patterns as a function of Ca and θ are shown

in Fig. 6.5. Based on the observed data, transitions between the three known displacement

regions can be identified, in a similar manner to how Zhang et al. [27] identified the existence

140

of different displacement regions for a range of Ca and viscosity ratios. Fig. 6.5 depicts the

transition between different regions in terms of Ca and θ.

θ

Ca

0 50 100 15010

­4

10­3

10­2

10­1

Stable displacement

Cap

illary

fin

ge

ring

Viscuous fingering

Figure 6.5: Displacement patterns at varying Ca and θ. Above a certain Ca threshold onlyviscous fingering is visible for the parameters studied in this work.

As seen in Fig. 6.5, as the viscosity ratio is held constant at one, all cases of Ca > 0.1

represent examples of viscous fingering, with little difference observed between media with

different hydrophobicity. As Ca drops and θ begins to increase, capillary fingering behaviour

is observed. Stable fronts are generally characterized by low Ca and hydrophilic obstacles.

6.4.3 Displacement Patterns

Here we demonstrate the progression of the front through the porous media described in the

previous section, for Ca = 10−3. Figs. 6.6, 6.7 and 6.8 depict snapshots of the fluid injection

141

into the porous media for contact angles of θ = 30, θ = 90 and θ = 150 respectively.

Figure 6.6: Injection simulation for θ = 30, with Re = 5 and Ca = 1× 10−3.

Figure 6.7: Injection simulation for θ = 90, with Re = 5 and Ca = 1× 10−3.

The transition from stable to unstable displacement is clearly visible. At small contact

angles, the front advances through nearly all available channels in part due to capillary rise

action. As the obstacles become increasingly hydrophobic, the adverse capillary pressure gra-

dient causes fingers to form, as well as pockets of defending fluid which are never displaced.

142

Figure 6.8: Injection simulation for θ = 150, with Re = 5 and Ca = 1× 10−3.

The presence of Haines jumps can be detected from the pressure differential across the porous

media, since occurrences are accompanied by sudden rapid decreases in local capillary pressure.

The pressure signals for various Ca and θ are shown in Fig. 6.9.

At low Ca, the dynamics of the flow are dominated primarily by viscous forces. As a

result, the average pressure gradient remains relatively constant, and varies only slightly as

the flow establishes a steady state. In the viscous regime, the dynamics of the interface are

nearly independent of contact angle. As Ca is lowered, two main things are observed to occur.

First, the pressure differential needed to drive the flow increases with θ. For the hydrophobic

obstacles, a high mean curvature interface forms as it travels through a throat, resulting in

a large capillary force acting in the direction opposite to the flow. For the more hydrophilic

obstacles, the direction of the curvature is reversed, which acts to expedite the progression of

the interface through the throat. The second observation is that due to the presence of Haines

jumps and capillary rise, the intensity of pressure fluctuations increases, and the permeability

of the porous media can be seen to continually decrease only when capillary fingering is present.

The degree to which capillary forces play a role in the flow dynamics can also be depicted

by examining the front propagation. In Fig. 6.10, the front displacement for different Ca at

various θ is shown.

143

T*

∆p*

0 0.2 0.4 0.6 0.8 10

100

200

300

(a)

T*∆p*

0 0.2 0.4 0.6 0.8 10

100

200

300

(b)

T*

∆p*

0 0.2 0.4 0.6 0.8 10

100

200

300

(c)

Figure 6.9: Driving pressure (∆p∗ = ∆pDµ|U| ) across the porous media domain, for (a) Ca = 10−1,

(b) Ca = 10−2 and (c) Ca = 10−3, for θ = 30 ( ), θ = 90 ( ) and θ = 150 ( ). Similarto [29], interface mergers/pore-filling events can be identified by rapid changes in the pressuresignal.

T*

X*

0 0.5 10.5

1

1.5

2

2.5

3

(a)

T*

X*

0 0.5 10.5

1

1.5

2

2.5

3

(b)

T*

X*

0 0.5 10.5

1

1.5

2

2.5

3

(c)

Figure 6.10: Comparison of front location of the invading fluid in the longitudinal direction,for (a) θ = 30, (b) θ = 90 and (c) θ = 150, with Ca = 10−1 ( ), Ca = 10−2 ( ) andCa = 10−3 ( ). As the capillary number decreases, the movement of the front becomesincreasingly dominated by capillary rise between pores and Haines jumps.

The dominance of capillary forces at low Ca manifests itself primarily through the unsteady

propagation of the front. This is due to high-frequency front oscillations resulting from both

capillary rise and Haines jumps, and lower frequency oscillations that are the result of the

formation of fingers. Additionally, some numerical artifacts related to variations in the contact

line velocity and the generation of spurious currents may also contribute to the pressure oscilla-

144

T*

S

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a)

T*S

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(b)

T*

S

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(c)

Figure 6.11: Displacement fraction S of the invading fluid with respect to the pore volume,for (a) Ca = 10−1, (b) Ca = 10−2 and (c) Ca = 10−3, with θ = 30 ( ), θ = 90 ( ) andθ = 150 ( ). In (a), saturation takes longer due to the presence of capillary fingering and theformation of droplets. The kinks in (b) and (c) are due to residual fluid pockets which are notdisplaced.

tions. Viscous regimes are characterized by a relatively smooth propagation of the front when

compared to capillary-driven regimes.

Examining the displacement fraction, S, can yield useful information regarding the different

displacement regimes. The displacement fraction S for the viscous fingering regime, Fig. 6.11,

shows a gradual increase in S up to Smax, where Smax denotes the displacement fraction after

steady state is obtained. In the viscous regime, the shear forces cause the interface to form

long fingers which primarily follow the local velocity field. As a result, the interface does not

immediately make contact with the surfaces of obstacles and form contact lines, which act to

fill the pores normal to the flow direction through capillary action. As a result, the time scale

on which the throats and pores normal to the flow direction are filled is larger when a contact

line does not form, as the interface will only fill pores at a rate governed by the flux through

pores themselves.

As Ca decreases, increasing the dominance of surface tension, and therefore capillary pres-

sure, the interface is forced to expand in the transverse direction and fill more pores regardless

of the pore orientation with respect to the flow. This effect can be seen in Figs. 6.11b and

6.11c, where at lower Ca, a more pronounced initial linear displacement phase is observed. In

contrast to Fig. 6.11a, however, the linear displacement phase can be seen to end abruptly. This

145

is due to the sudden formation of pockets of the defending fluid resulting from Haines jumps.

Only in the case of θ = 30 is full displacement observed, since this regime represents a fully

stable front which displaces all of the defending fluid.

6.5 Conclusions

This work presents an extensive computational study of the dynamics of a two-phase flow in

a porous media model. Direct numerical simulations provide an enhanced understanding of

the subtle and complex competition between capillarity and wetting effects in a porous media

consisting of two immiscible fluids displacing in a two-dimensional micro-geometry filled with

randomly sized and randomly distributed cylinders. We show the transition from a stable front

to a patterned displacement, with a focus on the wetting features of the porous media. The

study provides detailed insight into various scenarios as we change Ca and θ, while keeping

the viscosity ratio fixed. We leave the study of the viscosity ratio effect to future work. We

also study the Haines jump, namely rapid pore-scale displacement. Our simulations reveal the

characteristic time and length scales of the Haines jump. Our numerical analysis of the Haines

jump in a simple pore configuration, where cylinders of equal size are placed at the vertices of

equilateral triangles, provide further insight into the effect of the contact angle at which the

Haines jump is predicted. Finally, although we do not address issues related to the moving

contact line problem [30], our study does provide as complete a phase diagram as is available

in the literature. An extension of this work should consider the effects of slip, and validate the

results with experimental measurements.

Acknowledgements

We gratefully acknowledge helpful discussions with S. Zaleski. S. A. acknowledges the support

by the Petroleum Research Fund PRF-59641-ND.

146

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150

Chapter 7

Conclusions

7.1 Concluding Remarks

The interactions between fluid interfaces, contact lines and solid bodies are an important area

of study for a wide range of practical flows seen in industrial settings. These phenomena are

difficult to model, as they can involve fluid-stucture interactions in the case of moving bodies,

or complex domains in the case of porous media. The immersed boundary method offers a

promising approach to modelling such flows, as it eliminates the generation of traditional body-

fitted meshes all together.

In this work, new strategies for coupling IBMs with a multiphase VOF solver, complete

with surface tension and contact line phenomena, was presented. The efficacy of the proposed

algorithms was demonstrated for a wide range of test cases, ranging from circular droplets

sitting on cylinders, to the behaviour of particles at fluid-fluid interfaces, to the modelling of

porous media with varying hydrophobicity. The proposed schemes make no assumptions about

the methodology used to advect the volume fractions or to compute the surface tension, and

thus in principle are compatible with most standard interface advection and surface tension

computation schemes found in the literature.

In Chapter 4, a method based on the direct-forcing ghost-cell approach of Mohd-Yusof [1]

was presented. In this approach, the fluid and solid domains are treated as separate enti-

ties, with Eq. (2.1) and (2.2) solved in the fluid domain only. Contact lines were imposed on

the boundary through a dual approach, which consisted of extending the VOF field into the

151

boundary itself, and computing a modified surface tension force using a strategy inspired by

the CELESTE scheme of Denner and van Wachem [2]. The ability of the algorithm to recover

the equilibrium configuration of circular droplets on both flat and curved surfaces was demon-

strated. Several applications were also investigated, which included circular droplet impacts

onto cylinders, and the penetration of a circular droplet into a porous medium.

In Chapter 5, a method was proposed for simulating moving boundaries at fluid-fluid in-

terfaces. This approach differed from the one presented in Chapter 4 in that the fluid and

solid domains were not treated as separate, meaning Eq. (2.1) and (2.2) are solved in both the

fluid and solid domains. The IBM was a new indirect forcing approach which uses a series

of bi-quadratic functions to reconstruct the velocity field around the boundary surface, which

are then used to compute a body force term in the momentum equation. This approach is

much more suitable for moving boundary problems than the direct forcing ghost-cell approach.

Similar to the approach described in chapter 4, contact lines were included by both extending

the VOF field into the IB itself, as well as computing a modified interface curvature in the near-

boundary region. A number of test cases were studied, including axisymmetric simulations of

a spherical droplet situated on top of a solid sphere, as well as fluid-structure interaction prob-

lems such as the sinking of a horizontal cylinder, impacts of super-hydrophobic spheres, and

the formation of circular particle rafts at a fluid-fluid interface. Good agreement with available

experimental literature was demonstrated.

In Chapter 6, an application of the model proposed in Chapter 5 to the simulation of porous

media injection was investigated. The behaviour of a fluid interface in porous media injection

has been studied previously for a wide variety of physical parameters [3–5]. In this work, direct

numerical simulations using an IBM were used to investigate the effect of the contact angle and

Ca on the behaviour of the interface. The existence of different displacement regimes, which

includes stable displacement, viscous fingering and capillary fingering were identified for various

combinations of contact angle and Ca. Additionally, the effect of the hydrophobicity of the

obstacles on the permeability of the medium was shown. The effect of Haines jumps on the

displacement efficiency was also shown, which results in rapid pore-scale displacement, which

is accompanied by a sudden drop in capillary pressure.

152

7.2 Summary of Contributions

The main contributions of this work are the following:

• A new, fully-parallel CFD code for simulating interfacial flows with immersed boundaries

was created, and is publicly available at https://github.com/obrienadam/Phase.

• Coupled VOF advection algorithms with multiple different types of immersed boundaries

• A contact angle boundary condition for direct forcing ghost-cell static immersed bound-

aries was developed and validated.

• A new indirect forcing IBM for moving boundary problems was developed and validated,

complete with contact line boundary conditions.

• Several applications were investigated, most notably of which included the investigation

of front propagation in a randomly generated porous medium of varying hydrophobicity.

7.3 Future Work

A number of possible extensions to the current work are possible. In order to obtain more

accurate results for moving contact line problems, a mesh dependent contact angle similar in

nature to the one proposed by Afkhami et al. [6] should be integrated into the current models.

Most of these models rely on asymptotic solutions to the Cox equations [7], which can become

undefined at higher Ca. Therefore, consideration should be given to how to implement mesh

dependent contact angle boundary conditions for a wide range of Ca, since contact line velocity

for moving IBM problems may not be known a priori .

Another extension of the current work is to implement the models in a fully 3D framework.

To the best of our knowledge, no examples exist of moving IBM implementations with contact

angles. In principle, extension of the code to 3D should be relatively straightforward. The non-

existence of a unique unit tangent vector on the IB surface means that information regarding

the gradient of the VOF field would need to be used to perform the contact line extension.

Additionally, hexahedral cells in 3D contain 27 face, edge and vertex neighbours, as opposed to

153

just 8 face and vertex neighbours in 2D; thus the velocity reconstruction in Chapter 5 would

need to be investigated.

Additionally, only rudimentary consideration to solid body collisions was given in the present

work. For low Re regimes, viscous forces dissipate most of the energy of bodies as they collide,

thus collisions behave in a fairly inelastic manner. At higher Re, correct modelling of both

the normal impact forces and tangential friction forces between bodies becomes increasingly

important [8].

Finally, in this work, only simple geometries were considered. Real particulate flows would

involve a wide range of particle shapes, which in general will also be non-convex. This could

potentially complicate the inclusion of the IB for both direct and indirect forcing approaches.

Additionally, the extension of contact lines into the boundary itself might be more difficult,

as the stencil might intersect the body more than once due to its non-convex shape. Collision

detection between non-convex shapes also becomes more complex, however this has received

some attention previously in the literature [9–11].

154

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