Development of a novel immersed boundary lattice boltzmann method and its applications

DEVELOPMET OF A OVEL IMMERSED
BOUDARY-LATTICE BOLTZMA METHOD AD
ITS APPLICATIOS







WU JIE
(B. Eng., M. Eng., anjing University of Aeronautics and Astronautics, China)







A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMET OF MECHAICAL EGIEERIG
ATIOAL UIVERSITY OF SIGAPORE
2010

i
ACKOWLEDGEMETS
I wish to express my deepest gratitude to my supervisor, Professor Shu Chang, for his
invaluable guidance, supervision, patience and support throughout my Ph.D. study.

In addition, I would like to express my appreciation to the National University of
Singapore for giving me a research scholarship and an opportunity to do this program
at Department of Mechanical Engineering. It provides excellent conditions for me to
complete my research work smoothly. I also wish to thank all the staff members in the
Fluid Mechanics Laboratory for their valuable assistance.
My sincere appreciation will go to my dear family. Their love, concern, support and
continuous encouragement help me with tremendous confidence in solving the
problems in my study and life.
Finally, I would like to thank all my friends who have helped me in different ways
during my whole period of study in NUS.



WU JIE




ii
TABLE OF COTETS
ACKNOWLEDGEMENTS i
TABLE OF CONTENTS ii
SUMMARY viii
LIST OF TABLES x
LIST OF FIGURES xii
NOMENCLATURE xix

Chapter 1 Introduction 1
1.1 Background 1
1.2 Non-Boundary Conforming Method 3

1.2.1 Sharp interface approach 4
1.2.2 Diffuse interface approach 6
1.2.2.1 Immersed boundary method 7
1.2.2.2 Force calculation in IBM 8
1.2.2.3 Advantages and disadvantages of IBM 10
1.3 Lattice Boltzmann Method 12
1.3.1 Features of LBM 12
1.3.2 Historic development of LBM 13
1.3.3 Implementation of boundary conditions in LBM 15
1.3.4 Some efforts on improvement of computational efficiency 17
1.3.5 Applications of LBM with curved boundary and/or 17
iii
non-uniform mesh
1.3.6 Immersed boundary-lattice Boltzmann method (IB-LBM) 20
1.4 Objectives of The Thesis 21
1.5 Organization of The Thesis 23
Chapter 2 Development of Efficient Lattice Boltzmann Method on Non-Uniform 25
Cartesian Mesh
2.1 Standard LBM 26
2.2 Taylor Series Expansion and Least Squares-base Lattice Boltzmann 30
Method
2.3 Efficient LBM on Non-Uniform Cartesian Mesh 33
2.4 Accuracy Analysis of Present LBM 39
2.5 Numerical Tests 41
2.6 Concluding Remarks 43
Chapter 3 Boundary Condition-enforced Immersed Boundary-Lattice Boltzmann 51
Method
3.1 Immersed Boundary Method 52
3.2 Conventional IB-LBM 53
3.2.1 Combination of IBM and LBM 54

3.2.2 Penalty force model in IB-LBM 54
3.2.3 Direct forcing model in IB-LBM 56
3.2.4 Momentum exchange model in IB-LBM 58
iv
3.2.5 Drawback of conventional IB-LBM 59
3.3 Boundary Condition-enforced IB-LBM 60
3.4 Numerical Examples 68
3.4.1 Numerical test of overall accuracy 68
3.4.2 Flows over an array of circular cylinders placed at the middle of 70
straight channel
3.4.3 Flows over a circular cylinder 71
3.4.4 Flows over a pair of circular cylinders 75
3.4.4.1 Side-by side arrangement 76
3.4.4.2 Tandem arrangement 77
3.4.5 Flows over a NACA0012 airfoil 79
3.5 Concluding Remarks 81
Chapter 4 Application of New IB-LBM to Simulate Two-dimensional 105
Moving Boundary and Particulate Flow Problems
4.1 Brief Review of Boundary Condition-enforced IB-LBM 106
4.2 Simulation of Flows around Moving Boundaries 106
4.2.1 Brief review on moving boundary problems and computational 106
sequence
4.2.2 Numerical results and discussion 109
4.2.2.1 Flow over a moving circular cylinder 109
4.2.2.2 Flows over a rotationally oscillating cylinder 111
4.2.2.3 Unsteady flows at low Reynolds number flapping flight 115
v
4.2.2.4 Flows over a flapping flexible airfoil 117
4.3 Simulation of Particulate Flows 120
4.3.1 Brief review on the study of particulate flow problems 120

4.3.2 Force, torque calculation on the particle and computational 124
sequence
4.3.3 Numerical results and discussion 126
4.3.3.1 A moving neutrally buoyant particle in linear shear flow 126
4.3.3.2 Particle sedimentation in viscous fluid 128
A. One particle sedimentation 128
B. Two particles sedimentation 129
4.3.3.3 Particle suspension in a 2D symmetric stenotic artery 131
A. One particle passes the stenosis throat with b = 1.75d 131
B. Two particles pass the stenosis throat with b = 1.75d 132
4.4 Concluding Remarks 133
Chapter 5 Application of New IB-LBM to Study Flows over a Stationary 158
Circular Cylinder with a Flapping Plate
5.1 Brief Review on Flow Wake behind a Bluff Body 158
5.2 Configuration of Problem and Numerical Validation 160
5.2.1 Problem definition 160
5.2.2 Numerical validation 161
5.3 Numerical Study of Flows over a Stationary Circular Cylinder with 162
a Flapping Plate
vi
5.3.1 Flow patterns due to flapping plate 162
5.3.1.1 Effect of flapping frequency 162
5.3.1.2 Effect of flapping amplitude 164
5.3.2 Drag force due to flapping plate 165
5.3.3 Near-wake structure 167
5.3.4 Vorticity control 170
5.4 Concluding Remarks 174
Chapter 6 Extension of New IB-LBM to Simulate Three-dimensional Flows 186
around Stationary/Moving Objects
6.1 Three-dimensional Boundary Condition-enforced IB-LBM 186

6.2 Efficient Three-dimensional LBM Solver on Non-Uniform Cartesian 189
Mesh
6.3 Computational Sequence 190
6.4 Numerical Simulation of Flows around Stationary Objects 191
6.4.1 Flows over a stationary sphere 191
6.4.2 Flows over a torus with small aspect ratio 195
6.5 Numerical Simulation of Flows around Moving Objects 198
6.5.1 Flows around a rotating sphere 199
6.5.2 Fish swimming 202
6.5.3 Dragonfly flight 206
6.6 Concluding Remarks 208
vii
Chapter 7 Conclusions and Recommendations 230
7.1 Conclusions 230
7.2 Recommendations 234
References 236

viii
Summary
In recent years, the immersed boundary method (IBM) has been developed into a
popular numerical technique in the community of computational fluid dynamics
(CFD). As a successful example of non-boundary conforming methods, the Cartesian
mesh is utilized for resolving flow field in IBM. The effect of boundary is replaced by
the body force density which influences flow phase around the boundary. The
governing equations with this force density are solved on the whole computational
domain including the exterior and interior of the boundary. On the other hand, as an
alternative CFD tool, the lattice Boltzmann method (LBM) has gained wide range
applications recently. Since the Cartesian mesh is also employed in LBM, an efficient

solver can be generated by combining IBM with LBM, which is called IB-LBM.
Some efforts have been made in this aspect and the achievement is obvious. However,
there are still some shortcomings in this newly developed approach. In this work, two
major improvements were made.
Firstly, a new version of IB-LBM was proposed in order to strictly satisfy the
non-slip boundary condition. In the conventional IB-LBM, the non-slip boundary
condition is not enforced, and is only approximately satisfied at the converged state.
As a consequence, the accuracy of solution is reduced, and the situation of streamline
penetration to solid boundary is present. To overcome this drawback, a boundary
condition-enforced IB-LBM was developed. Different from the conventional IB-LBM
in which the body force is computed in advance, the unknown body force is employed
ix
in present method. Such force is resolved by enforcing the non-slip boundary
condition. Applying the developed approach, the two-dimensional (2D) stationary and
moving boundary flow problems, as well as particulate flow problems, were simulated.
Since the non-slip boundary condition is enforced, no flow penetration happens and
the accuracy of resolution is improved. All the obtained numerical results are
compared well with previous experimental and numerical results.
In the application of IB-LBM, the non-uniform mesh is usually employed in
order to improve the computational efficiency. To apply LBM on the non-uniform
mesh, many variants of LBM can be chosen. A simple way is to use Taylor series
expansion and least squared-based LBM (TLLBM). Its final form is an algebraic
formulation, in which the coefficients only depend on the coordinates of mesh points
and lattice velocity. As compared to the standard LBM, the drawback of TLLBM is
that additional memory is required to store the coefficients. Due to the limitation of
virtual memory, it is not easy to apply TLLBM in three-dimensional (3D) simulations.
To overcome this difficulty, an efficient LBM solver based on the one-dimensional
interpolation was developed. As compared to TLLBM, much less coefficients are
calculated. Combing with this efficient LBM solver, the new IB-LBM was easily
extended to 3D simulation. The 3D flows around complex stationary and moving

boundaries were simulated. The obtained numerical results are agreed well with the
results and findings in the literature.



x
LIST OF TABLES
Table 3.1 Comparison of forces
c
f
and
w
f
for flows over an array of 83
circular cylinders placed at the middle of a straight channel
Table 3.2 Comparison of drag coefficient, length of bubbles and separation 84
angle for flows over a circular cylinder with previous studies
Table 3.3 Comparison of drag coefficient, lift coefficient and Strouhal 85
number for flows over a cylinder with previous studies
Table 3.4 Comparison of drag coefficient, lift coefficient and Strouhal 86
number for flows over a pair of side-by-side cylinders at Re = 100
with previous studies
Table 3.5 Comparison of time-averaged drag coefficient, lift coefficient 86
and Strouhal number for flows over a pair of side-by-side
cylinders at Re = 200 with previous studies
Table 3.6 Comparison of drag coefficient, lift coefficient and Strouhal 87
number for flows over a pair of tandem cylinders at Re = 100
with previous studies

Table 5.1 Parametric study of flows over a stationary cylinder with a flapping 177

plate at Re = 100 for
0
0
10
θ
= and
0.2
f
St =

Table 6.1 Comparison of drag coefficients for flows over a sphere at Re = 100 210
and 200
xi
Table 6.2 Drag coefficients of flows over a torus 210

xii
LIST OF FIGURES
Figure 2.1 D2Q9 lattice velocity model 46
Figure 2.2 D3Q15 lattice velocity model 46
Figure 2.3 Schematic diagram of TLLBM, 2D case 47
Figure 2.4 Sketch of local interpolation scheme for LBM, 2D case 47

Figure 2.5 Comparison of velocity profiles for 2D lid-driven cavity flow 48
simulation
Figure 2.6 Comparison of velocity profiles on the plane of y = 0.5 for 49
3D lid-driven cavity flow simulation
Figure 2.7 Comparison of consumed CPU time for lid-driven cavity flow 50

Figure 3.1 Immersed boundary illustration 88
Figure 3.2 Configuration of boundary points and their surrounding fluid points 88
Figure 3.3 Convergence of numerical error versus mesh spacing for decaying 89
vortex problem
Figure 3.4 Streamlines for flows over a cylinder at Re = 20 and 40 89
Figure 3.5 Flow field for flows over a cylinder at Re = 100 and 200 90
Figure 3.6 Evolution of drag and lift coefficients for flows over a cylinder at 91
Re = 100 and 200
Figure 3.7 Geometrical configuration of flows over a pair of circular cylinders 92
Figure 3.8 Vorticity contours and streamlines for flows over a pair of side-by- 92
side cylinders (T = 1.5D) at Re = 100 and 200
xiii
Figure 3.9 Evolution of drag and lift coefficients for flows over a pair of side- 93
by-side cylinders (T = 1.5D) at Re = 100 and 200
Figure 3.10 Vorticity contours and streamlines for flow over a pair of side-by- 94
side cylinders (T = 3D) at Re = 100 in a cycle
Figure 3.11 Vorticity contours and streamlines for flow over a pair of side-by- 95
side cylinders (T = 3D) at Re = 200 in a cycle
Figure 3.12 Evolution of drag and lift coefficients for flows over a pair of side- 96
by-side cylinders (T = 3D) at Re = 100 and 200
Figure 3.13 Vorticity contours and streamlines for flow over a pair of tandem 97
cylinders (L = 2.5D) at Re = 100
Figure 3.14 Vorticity contours and streamlines for flow over a pair of tandem 98
cylinders (L = 2.5D) at Re = 200 in a cycle

Figure 3.15 Evolution of drag and lift coefficients for flow over a pair of tandem 99
cylinders (L = 2.5D) at Re = 200
Figure 3.16 Vorticity contours and streamlines for flow over a pair of tandem 100
cylinders (L = 5.5D) at Re = 100 in a cycle
Figure 3.17 Vorticity contours and streamlines for flow over a pair of tandem 101
cylinders (L = 5.5D) at Re = 200 in a cycle
Figure 3.18 Evolution of drag and lift coefficients for flows over a pair of tandem 102
cylinders (L = 5.5D) at Re = 100 and 200
Figure 3.19 Pressure contours and streamlines for flow over a NACA0012 103
airfoil at Re = 500
xiv
Figure 3.20 Distribution of pressure coefficient along the boundary of 103
NACA0012 airfoil at Re = 500
Figure 3.21 Vorticity contours and streamlines for flow over a NACA0012 104
airfoil at Re = 1000 and AOA = 10
°
in a cycle

Figure 4.1 Streamlines for moving and stationary cylinder cases at Re = 40 135
Figure 4.2 Vorticity distribution on the surface of cylinder at Re = 40 for 136
moving and stationary cylinder cases
Figure 4.3 Pressure profile on the surface of cylinder at Re = 40 for 136
moving and stationary cylinder cases
Figure 4.4 CPU time required for stationary and moving cylinder cases 137
at Re = 40
Figure 4.5 Comparison of vorticity contours for flow over a rotationally 137
oscillating cylinder when
=
f n
St St

at Re = 100
Figure 4.6 Vorticity contours at Re = 100 for flows around a rotationally 138
oscillating cylinder with different Strouhal numbers
Figure 4.7 Time histories of the lift and drag coefficients for two non lock-on 139
cases in rotationally oscillating cylinder simulation
Figure 4.8 Variations of drag and lift coefficients versus
f
St
in rotationally 140
oscillating cylinder simulation
Figure 4.9 Vorticity contours of a flapping wing in one cycle for
φ
= π
4
141
Figure 4.10 Time histories of lift and drag coefficients of a flapping wing 142
xv
for
φ
= π
4

Figure 4.11 Vorticity contours of a flapping wing in one cycle for
φ
−
= π
4
143
Figure 4.12 Time histories of lift and drag coefficients of a flapping wing 144
for

φ
−
= π
4

Figure 4.13 Plunge and deflection motion of a single flexible airfoil 145
Figure 4.14 Instantaneous vorticity contours for flapping airfoil in one cycle 147
Figure 4.15 Evolution of drag coefficients for rigid and flexible airfoils 147
Figure 4.16 The pressure contours for flapping airfoil in one cycle 149
Figure 4.17 Evolution of lift coefficients for rigid and flexible airfoils 149
Figure 4.18 Evolution of drag coefficients with different frequencies 150
Figure 4.19 Effect of flapping frequency on thrust power coefficient 150
Figure 4.20 Schematic diagram of a neutrally buoyant particle in a linear 151
shear flow
Figure 4.21 Comparison of lateral migration of particle with previous data 151
Figure 4.22 Comparison of particle translational velocities with previous data 152
Figure 4.23 Instantaneous vorticity contours of one particle sedimentation at 152
different time stages
Figure 4.24 Time histories of some quantities 153
Figure 4.25 Sedimentation of two particles in a channel at different time stages 154
Figure 4.26 Instantaneous vorticity contours of two particles sedimentation at 154
different time stages
Figure 4.27 Transverse coordinates of two particle centers 155
xvi
Figure 4.28 Longitudinal coordinates of two particle centers 155
Figure 4.29 Schematic diagram of the planar channel with stenosis 156
Figure 4.30 Snapshots of the positions of a particle 156
Figure 4.31 Velocities of the particle with respect to time 156
Figure 4.32 Trajectories of two particles symmetric to the centerline initially 157
Figure 4.33 Trajectories of two particles with the initial position asymmetric 157

to the centerline with asymmetry d/4000
Figure 4.34 Enlarged part of x- direction coordinates with respect to the time 157
when two particles pass through the throat

Figure 5.1 Flow over a stationary cylinder with a flapping plate 177
Figure 5.2 Flow patterns vary with flapping frequency at constant flapping 178
amplitude
0
0
20
θ
=
Figure 5.3 Flow patterns vary with flapping amplitude at constant flapping 179
frequency
0.4
f
St =
Figure 5.4 Variations of drag coefficients on the cylinder 180
Figure 5.5 Variations of overall drag coefficients on the cylinder and plate 180
Figure 5.6 Schematic diagram of vorticity distributions 181
Figure 5.7 Evolution of near-wake structure as a function of flapping 182
frequency at constant flapping amplitude
0
0
20
θ
=
Figure 5.8 Evolution of near-wake structure in a flapping period at
0
0

20
θ
= 183
and
0.2
f
St =
xvii
Figure 5.9 Evolution of near-wake structure in a flapping period at
0
0
20
θ
= 184
and
0.4
f
St =
Figure 5.10 Drag coefficient and corresponding flapping angle for destructive 185
interaction

Figure 6.1 Streamlines at the x-y plane for flows over a sphere at steady 211
axisymmetric state
Figure 6.2 Comparison of recirculation length
s
L
for flows over a sphere 212
Figure 6.3 Streamlines for flow over a sphere at Re = 250 212
Figure 6.4 Streamlines for flow over a sphere at Re = 300 213
Figure 6.5 3D vortical structures of sphere for planar symmetric flows 214

Figure 6.6 Dimensions of the torus 214
Figure 6.7 Streamlines for flows over a torus with Ar = 0.5 215
Figure 6.8 3D vortical structures behind torus without hole 216
Figure 6.9 Streamlines for flows over a torus with Ar = 2 217
Figure 6.10 3D vortical structures behind torus with a hole 218
Figure 6.11 Streamlines for flow over a rotating sphere at Re = 300 with 219
ω
∗
= 0.1 in a cycle
Figure 6.12 Streamlines for flow over a rotating sphere at Re = 300 with 220
ω
∗
= 0.5 in a cycle
Figure 6.13 Time histories of the drag and lift coefficients for flows over 221
a rotating sphere at Re = 300
xviii
Figure 6.14 Phase diagram of
y
C
and
z
C
for flows over a rotating sphere 222
at Re = 300
Figure 6.15 3D vortical structures of rotating sphere at Re = 300 223
Figure 6.16 Computational geometric form of RoboTuna 224
Figure 6.17 Time histories of the force coefficient for fish swimming 224
at Re = 4000
Figure 6.18 Streamlines with vorticity contours for fish swimming 225
at Re = 4000

Figure 6.19 3D vortical structures for fish swimming at Re = 4000 226
Figure 6.20 Computational model of dragonfly 227
Figure 6.21 Sketch diagram for dragonfly flapping wing 227
Figure 6.22 Time histories of x-, y-, and z-direction force coefficients for 228
dragonfly flight at Re = 500
Figure 6.23 Evolution of pitching angle and lift coefficient at one hindwing 229
Figure 6.24 3D vortical structures for dragonfly flight at four different stages 229













xix
OMECLATURE
Roman Letters
d
C
Drag coefficient
l
C
Lift coefficient
p

C
Pressure coefficient
f
c
Force scale factor
s
c
Sound speed
D Diameter of circular cylinder
α
e
Lattice velocity vector
x
e
α
,
y
e
α
,
y
e
α
Lattice velocity
F
,
f
Force density vector
D
F

Drag force
L
F
Lift force
q
f
Vortex shedding frequency
f
α
Density distribution function
eq
f
α
Equilibrium density distribution function
g
α
Post-collision state of distribution function
h
Mesh spacing
k
Spring constant
s
L
Length of bubble
p
,
B
p
,
p

∞
Pressure
xx
Re
Reynolds number
St
,
f
St
,
n
St
Strouhal number
s
∆
Arc length of boundary element
T
Period
t
,
1
t
,
2
t
Physical time
t
δ
Time step
U

,
B
U
,
u
,
∗
u
Velocity vector
U
∞
Free stream velocity
u
,
v
,
w
Velocity
δ
u
,
B
δ
u
Velocity correction vector
w
α
Coefficients
X
,

B
X
,
x
,
1
x
Coordinate vector
x
,
y
,
z
Coordinate

Greek Letters
α
,
β
Lattice velocity direction
δ
Dirac delta function
ε
Small expansion parameter
d
ε
Stiffness scale factor
s
θ
Separation angle

µ
Dynamic viscosity
ξ
Displacement
xxi
(
)
0
Π
,
(
)
1
Π
Momentum flux tensor
ρ
Density
τ
Single relaxation parameter
φ
Phase difference
υ
Kinematic viscosity
Ω
Computation domain
Ω
Angular velocity vector
ω
,
B

ω
Vorticity

Abbreviations
2D Two-dimensional
3D Three-dimensional
ALE Arbitrary Lagrangian-Eulerian
DLM/FD Distributed Lagrange multiplier/fictitious domain
DKT Drafting, kissing and tumbling
IBM Immersed boundary method
IB-LBM Immersed boundary-lattice Boltzmann method
IIM Immersed interface method
LBE Lattice Boltzmann equation
LBM Lattice Boltzmann method
N-S Navier-Stokes
TLLBM Taylor series expansion- and least square-based LBM

1
Chapter 1
Introduction
1.1 Background
As a branch of fluid mechanics, the computational fluid dynamics (CFD) plays an
important role in the research and application of engineering problems. The basic
principle of CFD is to use numerical methods to get approximate solutions of the
governing equations for the flow field at discrete (mesh) points. The ultimate aim of
CFD is to understand the physical events by numerical methods which happen in the
fluid flows over and/or within specified objects.
The field of CFD has a wide range of applicability in engineering. It ranges from
the study of air flow around an airplane, wind past tall buildings, blood flow through
heart valves, to river sediment resuspension and transport, cell deformation and

transport, fish swimming, insect and bird flight and so on. Broadly speaking, such
problems can be generally divided into three groups: (1) the object does not move in
the fluid; (2) the object undergoes prescribed motion in the fluid; (3) the object moves
freely in the fluid. Among the three groups, the simulation of flows around moving
objects is still a challenging area in CFD.
The current numerical methods for simulating flows around moving objects can
Chapter 1 Introduction
2
be roughly classified into two major categories: (1) boundary conforming method; and
(2) non-boundary conforming method. For the boundary conforming method, the
body-fitted mesh is often used. This means that the boundary points coincide with
mesh points. Hence, the physical boundary condition can be imposed at the boundary
point directly. To simulate the flow over a moving object under the boundary
conforming framework, one choice is to use the time-dependent coordinate
transformation technique (Tsiveriotis and Brown, 1992; Mateescu et al. 1996;
Dimakopoulos and Tsamopoulos, 2003; Luo and Bewley, 2004). In this approach, the
physical domain is transformed into a computational domain. All the numerical
computations are easily performed in the computational domain. The most popular
boundary conforming method is perhaps the arbitrary Lagrangian-Eulerian (ALE)
approach (Hirt et al. 1974; Hu et al. 2001; Anderson et al. 2004; Chew et al. 2006). In
many applications, the unstructured finite element (FE) mesh is employed in the ALE
method. In this category, a scheme named space-time finite element method
(Tezduyar et al. 1992a, 1992b) can be viewed as the general form of ALE finite
element scheme. In the boundary conforming method, since the implementation of
boundary condition is straightforward, accurate solutions can be achieved. The main
difficulty of boundary conforming method is the complexity involved in regenerating
the mesh to conform to the object boundary at all times, especially when the boundary
surface is very complex. Therefore, it would be desirable to decouple the boundary
from the computational mesh which is used to solve governing equations of the fluid
flow. This motivates the development of non-boundary conforming method. In this

Chapter 1 Introduction
3
category, the flow field is solved on a fixed Cartesian mesh, and the boundary points
do not need to coincide with the mesh points. The effect of boundary on the flow field
is accounted through the proper treatment of the solution variables at the mesh points
or cells around the boundary. As compared to the boundary conforming method, the
non-boundary conforming method eliminates the requirement of tedious grid
adaptation, which makes the simulation of flows over complex/moving objects be
more straightforward.
The primary purpose of this thesis is to develop efficient computational
approaches in the category of non-boundary conforming method for accurate
simulation of flows over stationary and/or moving objects. In the present work, the
flow field is obtained by the lattice Boltzmann method (LBM). In the following, we
will firstly give a literature review on the non-boundary conforming method and the
LBM, and then describe the objectives of this research and layout of the thesis.

1.2 on-Boundary Conforming Method
Up to now, a plethora of prominent methods have been developed to handle flows
over objects based on the fixed Cartesian mesh. Unlike the boundary conforming
method which is fairly standardized, the non-boundary conforming method
demonstrates great differences in implementation details, depending on the problems
and the definition of boundary. In terms of treatment of boundary conditions, the