Development of immersed boundary methods for isothermal and thermal flows 1

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Chapter 1

Introduction

1.1 Background of computational fluid dynamics
Computational fluid dynamics, frequently known as CFD, is a branch of fluid
dynamics which uses numerical methods to predict problems including fluid
flows, heat transfers and other related phenomena. Presently, the great strides
in computers have driven CFD as an important alternative to expensive
experiments and bewildering theoretical fluid dynamics. Researchers and
engineers are strongly encouraged to rely on CFD for the analysis of fluid
dynamics-related problems and technologies.

The principle of CFD is to pursue an approximate numerical solution for the
governing equations of the flow field, i.e. Navier-Stokes (N-S) equations. The
general procedure for this includes: (1) mesh or grid generation – the fluid
region of interest is divided into a collection of finite cells or discrete points;
(2) discretization of governing equations – the Navier-Stokes equations, which
are generally partial differential equations, are discretized into the discrete
equations on the interior grids/cells by employing some appropriate numerical
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schemes; (3) implementation of boundary condition – the boundary conditions

on the boundary grids/cells are reconstructed, which, together with the
aforementioned discrete equations, form a set of well-defined algebraic
equation system; (4) solution of resultant equations – the set of algebraic
equations are then solved numerically at each cell or point to get numerical
solutions for the fluid domain. We can see clearly that the numerical solution
strongly depends on the grid generation process and the discretization method
for the governing equations.

Traditionally, body-fitted mesh is often used, in conjunction with the classical
finite difference (FD) and finite volume (FV) method. These traditional
body-fitted methods perform well and enjoy certain popularity in many areas
of scientific research and engineering analysis.

1.1.1 Limitations of traditional body-fitted method
Despite the good performance and popularity of the traditional body-fitted
methods, their wider applications have been limited due to the geometrical
complexities frequently encountered in flow problems. Many scientific and
engineering practices involve bodies with complex geometries, or objects
under moving and/or deformation, which would present considerable
computational difficulties for the body-fitted method. For example, mesh
generation of the computational domain, could be a very troublesome issue, if
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considering its significant impact on convergence rate, solution accuracy and
CPU time required. To overcome the difficulties associated with the
geometrical complexity, two techniques have been introduced: structured
curvilinear mesh for FD and FV methods and unstructured mesh for FV and
FE (finite element) methods.

Structured curvilinear mesh allows boundaries to be aligned with constant
coordinate lines and is capable of providing a good representation of
boundaries and surface boundary layers, simplifying the treatment of boundary
conditions and reducing the numerical “false-diffusion” errors, etc. The
construction of structured curvilinear mesh always resorts to the coordinate
transformation and mapping techniques which would transform a complex
physical domain into a rectangular computational domain. However, during
the projection process, a highly accurate method is required to calculate the
transformation Jacobian matrix. Otherwise, additional geometrical errors will
be introduced and the accuracy of the domain is thus degraded. Furthermore,
the coordinate transformation is problem-dependent and tedious. Even for
seemingly simple geometries, generating a good-quality body-fitted structured
mesh can always be an iterative process with a substantial amount of time, not
to mention more complicated ones.

In comparison, the unstructured mesh for FV and FE methods makes use of
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arbitrarily shaped polygons (such as triangles, quadrilaterals in two-dimension,
or tetrahedral, pyramids, prisms in three-dimension) and thus seems to offer
greater flexibility to fit the complex shape of the physical domain. Although
meshing effort can be saved by using the unstructured mesh, there is some
memory and CPU overhead for unstructured referencing since a list of
connectivity pattern which specifies how a given set of vertices make up
individual elements is required to be stored. In addition, the grid quality and
robustness can be aggravated with increasing complexity in the geometry. It is
also noted that the unstructured grid method originally emerged as a feasible

alternative to the structured grid technique for discretizing complex
geometries. However, owing to the inapplicability of powerful line/block
iteration and geometrical multi-grid techniques to unstructured grid,
unstructured grid methods are in general slower on a per-grid-point basis than
structured grid methods.

On the other hand, moving boundary problems pose an even greater challenge
to grid generation, especially when they are combined with geometrical
complexity. With the movement of bodies or objects, the physical fluid
domain changes continuously. In view of the body-fitted concept, the
grid/mesh should be moving correspondingly to conform to the configurations.
However, in most cases, the mesh deforms to such an excessive distortion that
the computation would break down. To avoid this, successive re-meshing of
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the domain is required. This inevitable grid/mesh regeneration is remarkably
expensive and unsatisfactory. Additionally, the solution variables need to be
projected from the old mesh to the new one after re-meshing. This
interpolation process not only brings forth heavy computational burden, but
also leads to undesirable degradations of solution accuracy, robustness and
stability.

1.1.2 The concept of non-body-conforming method
In the last two decades, a group of so-called non-body-conforming Cartesian
grid methods have been proposed, in an attempt to overcome the weakness of
the body-fitted grid methods. As its name implies, the non-body-conforming
methods are specially designed to eliminate the necessity of adapting the
underlying computational mesh to the physical configuration of the fluid

domain. One of the key advantages of non-body-conforming Cartesian grid
methods lies in time and human-labor savings on the mesh construction. Since
the Cartesian grid is generally utilized, the grid complexity is relieved from
the geometric complexity. For moving objects, there is no need for grid
re-generation at each time step. At the same time, the method retains most of
the favorable properties of structured grids such as easy application of
line/block iterative method and geometric multi-grid method. In this way, the
non-body-conforming Cartesian grid methods can tackle flows involving
complex geometries or moving boundaries with relative ease. That is why the
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non-body conforming method has become popular in recent years.

1.2 Non-body-conforming method
The introduction of non-body-conforming Cartesian grid method is credited to
Peskin who proposed an immersed boundary method (IBM) in 1972 when
studying the blood flow and cardiac mechanisms inside the human heart
(Peskin, 1972). Since then, more and more scholars have been attracted,
showing strong interests in improving the method and widening its application.
As a result, various non-body-conforming Cartesian grid methods have been
springing up in the last two decades. Generally, the non-body-conforming
method always takes a regular region, which may frequently be a rectangular
one, as its computational domain. The domain is sufficiently large to cover the
entire problem region inside it. The complex-geometric and/or moving bodies,
under such circumstances, are regarded as interfaces or boundaries immersed
in the domain.

As learnt from its name, the non-boundary-conforming Cartesian grid is not

aligned with the geometry of the physical domain. Therefore, imposing the
boundary conditions is not as straightforward as the traditional body-fitted
method. As a result, a procedure which is capable of incorporating the
boundary condition (or the effects of the boundary) into the overall algorithm
and, at the same time, does not affect the accuracy or significantly increase the
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computational cost, is definitely required. It is this challenging procedure that
distinguishes one method from the other. Based on whether the immersed
boundary is treated as an interface with a finite thickness or not, the existing
non-body-conforming Cartesian grid methods can be broadly classified into
two categories: sharp interface method and diffuse interface method. In the
sharp interface method, the boundary is viewed as a zero-thickness sharp
interface. The ghost-cell method, cut-cell method, immersed interface method
fall into this category. In the diffuse interface method, the effect of boundary is
smeared out across the interface to a thickness of the order of the mesh width.
The immersed boundary method mentioned above is among this category.

1.2.1 Sharp interface method
The sharp interface methods are capable of accurately capturing the solid
interfaces and enforcing the boundary conditions on them, at the expense of
complicated algorithms for accurate implementation of boundary conditions.
Some representatives like ghost cell method, cut cell method and immersed
interface method are reviewed in the following.

1.2.1.1 Ghost cell method
In the ghost cell method, the boundary conditions on the fluid-solid interface
are imposed through the flow variables at the “ghost-cells”, whose cell centers

are falling inside the solid region but having at least one neighboring fluid cell.
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Employing an appropriate local reconstruction scheme (interpolation or
extrapolation), the flow variable values of the ghost cells are calculated in
such a way that the prescribed boundary condition at the interface is satisfied.
Different reconstruction schemes such as linear, bilinear and quadratic ones
(Majumdar, 2001; Tseng & Ferziger 2003; Pan & Shen 2009) require different
reconstruction stencils, and their complexity determines the methodology
complexity. For example, a linear reconstruction model (Tseng & Ferziger
2003) can be employed, utilizing the projection point of the ghost cell on the
immersed boundary and two fluid points nearest to the projection point as the
stencil for extrapolation. However, when any of the two fluid points in the
stencil is too close to the interface, numerical instability will arise.
Furthermore, it is more likely to introduce spurious oscillations with more
stencil points. So the ghost cell method may be troubled by the robustness
issue associated with supporting stencils for the reconstruction scheme.

1.2.1.2 Cut-cell method
The cut-cell method is another typical Cartesian grid-based sharp interface
method. In the cut-cell method, a series of irregular truncated cut cells which
exist immediately adjacent to the boundary play the role of implementing the
boundary conditions. In practice, the truncated cut cells may be arbitrarily
small (especially for highly curved or complex boundary) and would lead to
severe numerical instability. To avoid an impractical time step size, a cell
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merging technique in which the cut cell was absorbed by an appropriately
selected neighboring cell is usually necessary (Ye et al. 1999; Chung 2006).
After the cut cells are reshaped, the governing equations are discretized in
these merged cells based on their actual shape. However, due to the various
manners the boundary may intersect with the background regular mesh,
numerous scenarios for shapes of the merged cut cells should be accounted for.
There is another difficulty which frequently disturbs the application of cut-cell
method – the presence of degenerate cut cells (Je et al., 2008). In
two-dimension, the degenerate cut-cells are those that (1) have more than two
intersection points with the boundary curve or (2) have more than one
intersection point with any cell face. Further, as pointed out by Mittal &
Iaccarino (2005), “successful implementation of the cut-cell method to
three-dimensional geometries has not yet been accomplished.”

1.2.1.3 Immersed interface method
The immersed interface method (IIM) was originally proposed by LeVeque &
Li (1994) for elliptic equations with discontinuous coefficients, and was later
extended to account for two-dimensional incompressible flows with interfaces
or immersed boundaries (LeVeque & Li, 1997; Li & Lai, 2001; Xu & Wang,
2006; Le et al. 2006). In practice, the existence of interfaces or immersed
boundaries may lead to jumps in pressure and in the derivatives of both
pressure and velocity at the interface/boundary. The basic principle of IIM for
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fluid dynamic problems is that the jump conditions in the flow variables
and/or their derivatives are explicitly incorporated into the difference
equations to achieve second or even higher order of accuracy. However, the

determination of jump conditions across the immersed boundary is not an easy
job at present. Firstly, they normally have a very complicated form even for
the simple membrane flow system. Secondly, the derivation of the jump
conditions always requires the immersed interface to be a closed structure, i.e.,
a closed curve in two-dimension or a closed surface in three-dimension (Xu &
Wang, 2006). The IIM is also troubled with the drawback that special
finite-difference stencils need to be particularly designed for the discretization
of Navier-Stokes equations near the immersed boundaries.

In summary, the success of any sharp interface Cartesian grid methods
depends strongly on how the boundary conditions are implemented and how
the discretization schemes are modified at the immersed boundary, which is
frequently accompanied by an iterative data reconstruction procedure and
elaborate efforts for special mesh treatment.

1.2.2 Diffuse interface method
It can be recognized that a common difficulty for various sharp interface
methods is the requirement of irregular stencils near the immersed boundary
for derivative approximation or data reconstruction scheme. Compared to the
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sharp interface Cartesian grid methods, the diffuse interface methods are
relieved from these troubles and relatively easy to implement. Representatives
are fictitious domain method and immersed boundary method.

1.2.2.1 Fictitious domain method
The fictitious domain method enforces the conditions on the immersed
boundaries in a weak form by means of Lagrangian multipliers (Glowinski et

al., 1994). Using the fictitious domain method, the physical solution in the
fluid domain is required to extend into the solid domain (frequently referred to
as the fictitious or artificial fluid domain) in a continuous manner (Glowinski
et al. 1994). For example, the fictitious fluid solution in Glowinskin et al.
(1995) and Yu et al. (2006) was directly enforced to be the same as the solid
solution. For problems where the solid bodies have prescribed motions, the
constraint together with corresponding Lagrangian multipliers only need to be
set along the fluid-solid interface, as suggested by Glowinski et al. (1997).
However, if the solid body motion is not given in advance but caused by the
hydrodynamic forces and torques, the constraint of rigid-body motion on the
fluid-solid interface alone is not enough. In this case, Glowinski (1999)
exploited a distributed Lagrangian multiplier-based fictitious domain method
(DLM/FD) in which the constraint of rigid-body motion was extended and
imposed on the fictitious fluids as well. Although successfully applied to
problems like particulate flows (Patankar et al. 2000; Yu et al. 2006), the
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Lagrangian multiplier in the fictitious domain method is normally calculated
implicitly from the rigid-body motion constraint. This implicit determination
causes DLM/FD method to suffer from expensive computations.

1.2.2.2 Immersed boundary method
A very popular and attractive diffuse interface method in last decades is the
immersed boundary method (IBM), which was developed by Peskin when he
studied blood flow in human heart (Peskin, 1972). In his work, the human
heart was modeled as an elastic membrane immersed in a rectangular flow
domain. He used a fixed Eulerian Cartesian mesh to describe the blood flow
and a set of elastic fibers (represented by a series of Lagrangian points which

can move and deform freely through the underlying Eulerian mesh) for the
heart motion. The interaction between the heart and blood flow was realized
through the introduction of Dirac delta function. Once the heart moves or
deforms, singular forces are generated along the heart wall. These singular
forces at the Lagrangian points are then spread to their surrounding Cartesian
Eulerian grids as body forces via a discrete delta function. The incompressible
Navier-Stokes equations with the additional body forces are then solved on the
entire domain including both the interior and exterior of the human heart.
After the velocity on the fixed Eulerian gird are calculated, the heart is
updated to its new shape and location according to the no-slip condition
between the blood flow and the heart wall. In this way, the coupling between
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the governing equation solver and the boundary condition implementation is
eliminated, and dynamically updating the geometry changes becomes
straightforward. As such, the solution to the whole system (blood flow + heart
motion) is easily yielded.

From the above illustration, it can be observed that the immersed boundary
method is conceptually independent of the spatial discretization and is simple
to implement in an existing Navier-Stokes solver. By modeling the immersed
boundaries as force sources, it can handle complex geometries easily without
any special mesh treatment, even for flexible boundaries undergoing a
complicated movement or shape variation. In fact, the method has proven to
be a versatile and successful tool for problems with complex geometries and
moving boundaries. In this regard, the immersed boundary method attracts our
attention and is studied in the present thesis. It should be noted that although
some methods (Deng et al. 2006; Choi et al. 2007; Zhang & Zheng 2007;

Paravento et al. 2008; Liao & Lin 2012; Noor et al. 2009; Ghias et al. 2007;
Chen et al. 2013; Mittal et al. 2008; etc.) in the literature are also claimed to be
the immersed boundary methods and introduce the momentum forces into the
governing equations to represent the effect of the immersed objects, they treat
the immersed boundary as a sharp one, which is quite different from Peskin’s
original method. These “so-called immersed boundary methods” are not real
immersed boundary methods and therefore fall out of the scope of the present
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thesis.

1.3 Brief review of Immersed boundary methods
The immersed boundary method has received great attention since being
published. Following Peskin’s pioneer contribution, abundant variations of the
method have come forth. Among them, some are devoted to the improvement
or refinement of the fluid solver while others concentrated on widening the
application fields of the method.

In general, the diverse immersed boundary methods based on the
Navier-Stokes solvers are established in two frameworks according to the
underlying form of Navier-Stokes equation utilized: pressure-velocity
formulation-based immersed boundary method, and stream function-vorticity
formulation-based immersed boundary method. Consequently, researches on
algorithm improvement or refinement have been proceeding along the two
directions.

1.3.1 Pressure-velocity formulation-based immersed boundary method
The pressure-velocity formulation-based immersed boundary method follows

Peskin’s original work, in which the body force term is explicitly incorporated
into the momentum equation to represent the effect of the immersed boundary.
The previously proposed methods reveal that the boundary/body force is
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introduced in order to approximate or enforce the no-slip condition on the
immersed boundaries. So, how to accurately evaluate the force term becomes
a critical issue for a successful implementation of the pressure-velocity
formulation-based IBM.

The calculation of body force depends on the characteristics of the immersed
boundary: elastic and rigid boundaries require different treatments. The
present thesis mainly focuses on objects with rigid boundary. The elastic
boundaries (Zhu & Peskin 2002; Kim & Peskin 2007; Francois & Shyy 2003;
Udaykumar et al.1997; Kim & Peskin 2006) can be handled perfectly by using
the constitutive law as what Peskin’s original study did (Peskin 2002).
However, utilizing the constitutive law to rigid boundaries would pose
problems (Mittal & Iaccarino, 2005). As a result, tremendous efforts have been
spent to tackle the issue of how to properly calculate the force density on the
solid boundaries. Several remarkable strategies are introduced in the
following.

Lai & Peskin (2000) proposed a penalty force scheme to calculate the singular
Lagrangian force density. They assumed that the boundary points of the
immersed object were being attached to their equilibrium positions by springs
with high stiffness. When the boundary deviates from the equilibrium location,
a restoring force will be generated according to the Hooke’s law so that the
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boundary points will stay close to the target boundary position.

Goldstein et al. (1993) generalized the above idea and provided a two-mode
feedback forcing scheme for the control of velocity condition at the immersed
boundary. This forcing term reflects the velocity difference between the
desired boundary value and the interpolated one, and behaves in a feedback
loop such that the boundary velocity remains close to the desired value. The
approach has been employed for the simulation of low Reynolds number
turbulent flows over a riblet-covered surface (Goldstein et al. 1995) and
self-propelled fish-like swimming (Wang et al. 2014).

Later, Mohd-Yusof (1997) suggested a forcing evaluation approach in which
the body force was directly derived from the transformed momentum equation.
The method is frequently termed the direct forcing method. Fadlun et al. (2000)
successfully applied the approach to large-eddy simulation of turbulent flow.
Although this force evaluation scheme was initially suggested in a sharp
interface method, Uhlmann (2005) has successfully generalized this idea into
Peskin’s immersed boundary method, by incorporating the regularized delta
function into the force calculation and spreading process.

Shu et al. (2007) discovered that most of the aforementioned methods cannot
accurately satisfy the no-slip boundary condition and penetration of some
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streamlines happens on the solid boundary. The penetration may degrade the

accuracy of the method and also the boundary force calculation. To remove
flow penetration through the solid boundary, a velocity correction approach
was proposed by Shu et al. (2007), where the velocity correction is made in
the vicinity of the boundary points such that the no-slip boundary condition is
directly enforced. Following the idea of Shu et al. (2007), Wu & Shu (2009)
recently proposed an implicit velocity correction-based immersed boundary
solver, in which the velocity correction is determined implicitly in a way that
the velocity at the boundary interpolated from the corrected velocity field
through the discrete delta function accurately satisfies the no-slip boundary
condition. It should be noted that although the immersed boundary solvers of
Shu et al. (2007) and Wu & Shu (2009) can guarantee the accurate satisfaction
of no-slip condition on the immersed boundary, the flow field in both works is
delivered by the Lattice Boltzmann method (LBM), whose computational
efficiency is always constrained by its intrinsic limitation of requiring special
uniform lattice, thus making it challenging to offer high reolution near a solid
body and/or include far-field boundary. In this regard, it would make good
sense to explore how to implement the velocity correction-based IBM into the
Navier-Stokes solver and how well it would perform.



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1.3.2 Stream function-vorticity formulation-based immersed boundary
method
Literature shows that almost all the established immersed boundary methods
are inherited from Peskin's original approach. Their common feature is to
introduce the momentum forcing terms (i.e., body force term) into the

governing equation to represent the effect of the immersed boundary. In this
regard, the IBM seems to be more effective under the framework of
velocity-pressure (primitive-variable) formulation. On the other hand, we have
to indicate that for incompressible flows, the primitive-variable formulation
suffers from some difficulties in the solution process (Shu & Wee, 2002). As
can be seen, the velocity components appear in both the momentum and
continuity equations, but the pressure only appears in the momentum equation.
When the momentum equations are used to compute the velocity components,
there is no guarantee that the obtained velocity components would satisfy the
continuity equation. In addition, there is no transport equation for computing
the pressure. This brings difficulty for coupling between the velocity field and
the pressure field (Chaviaropoulos & Ciannakoglou, 1996). To remove this
difficulty, special techniques are required, and a pressure Poisson equation is
usually introduced. The solution of elliptic Poisson equation often takes a lot
of computational time. Furthermore, it requires the use of staggered grid,
which could bring complexity in programming. In contrast, the stream
function-vorticity formulation of Navier-Stokes equations has been well
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recognized to be more efficient for 2D incompressible flows. In this form, the
momentum equations are combined and transformed into a transport equation
for vorticity

ω
, where the pressure gradient term disappears. This clearly
eliminates the critical issue of velocity-pressure coupling. Besides, the
continuity equation is automatically satisfied due to the introduction of stream
function

ϕ
, so the normal grid can be effectively utilized. Furthermore, two
variables (
,
ω
ϕ
) rather than three variables (
pvu ,,
) are solved. This would
reduce the computational effort.

To make a good use of the respective merits inherited from the stream
function-vorticity formulation and the IBM, it is desirable to develop a stream
function-vorticity formulation-based IBM solver. However, the basic idea of
original IBM is that the forcing terms are added to the momentum equations to
show the effect of immersed boundary. Obviously, this idea cannot be directly
incorporated into the stream function-vorticity formulation as the momentum
equations are differentiated and combined into a single equation for the
vorticity. Due to this difficulty, there is little work in the literature in this
development. To the best of our knowledge, Wang et al. (2009) made an effort
to study this problem. In their method, the vorticity-velocity formulation is
adopted and the spirit of Peskin's original IBM is directly applied. Both
velocity and vorticity are divided into two parts. One is the velocity and
vorticity without the influence of immersed boundary, and the other is the
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velocity and vorticity correction. In order to correct the velocity, the
multi-direct forcing scheme suggested by Wang et al. (2008) and Luo et al.

(2007) is utilized, which, as a matter of fact, is an iterative process to
approximately satisfy the no-slip condition at the immersed boundary. In the
solution process, the no-slip condition is not enforced. For vorticity correction,
source terms, which are derived from the momentum equations with forcing
terms by differentiating with respect to y (for x-momentum equation) and x
(for y-momentum equation), are added into the vorticity transport equation.
These terms are very complicated, and involve the first, second and third order
spatial derivatives. Numerical dicretization of these spatial derivatives brings
complexity into the computation. In view of simplifying the computational
complexity, an efficient stream function-vorticity formulation-based IBM
solver for simulating 2D incompressible viscous flows is desired.

1.3.3 Applications of immersed boundary method
The popularity of immersed boundary method has seen increased for flow
problems with complex or/and moving boundaries. Although being created to
study the fluid dynamics of cardiovascular physiology such as flow in heart
valves (McQueen & Peskin 2000), flow in arterioles (Arthurs et al. 1998),
deformation of red blood cells in a shear flow (Eggleton & Popel 1998), etc.,
the method has been applied to various biological flows including wave
propagation in the cochlea of the inner ear (Beyer Jr. 1992), thrust generation
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of swimming fish (Wu & Wang 2009) and lift production of flying insects
(Miller & Peskin, 2005). In addition, a variety of applications concerning
fluid-structure interaction (Sotiropoulos & Yang 2014), turbulent flow (Shun
et al. 2014), multiphase flow (Li et al. 2012) and multi-component flow (Du et
al. 2014) have also been reported. Examples may refer to parachutes inflation
(Kim & Peskin, 2009), particle sedimentation (Feng & Michaelides, 2004),

topological changes of wet foam (Kim & Seol 2013), flapping motion of
turbine blades (Kalitzin & Iaccarino, 2003), even flow around a walking
person (Choi et al. 2007).

Apart from the applications to fluid flows, there are some other studies with a
relevance to heat transfer problems. Due to considerable importance of heat
transfer in a wide range of engineering applications, such as convection in
buildings, solar energy systems, electronic cooling equipment, crystal growth
processes and nuclear reactor, a better understanding of the involved flow and
thermal physics in these phenomena has a fundamental significance to their
improvement. However, most of these thermal problems are suffering from
complex or moving configuration, which stimulates scholars to generalize the
traditional immersed boundary method and extend it to the thermal flow field.
Since the basic idea of IBM is that the effect of the immersed boundary on its
surrounding fluid is realized through introducing forcing terms in the
momentum equations, it has a difficulty to be directly applied to heat transfer
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problems since they also involve the energy equation. In fact, an inspection of
literatures shows that there is relatively little work on applying IBM to heat
transfer problems as compared to its application for fluid dynamic problems.
Nevertheless, there are some efforts to extend IBM for application to thermal
flow problems with Dirichlet-type boundary condition. Among notable
achievements, Zhang et al. (2008) extended their early work for fluid flows
(Zhang & Zheng 2007) to the heat-transfer problem. Similar to the
introduction of a forcing term into the momentum equations, they incorporated
a heating term in the energy equation to represent the virtual heat source. It is
evaluated from the difference between the given temperature and the

computed one at the Lagrangian point, which is then mapped back to the
Eulerian grid points using the same idea as in the traditional feedback-forcing
IBM (Saiki & Biringen 1996). Wang et al. (2009) investigated problems of
natural convection between concentric cylinders as well as forced convection
around a stationary circular cylinder using a multi-direct heat source scheme,
which is similar to the multi-direct forcing method (Wang et al. 2008)
proposed previously for isothermal flows. Young et al. (2009) combined the
direct-forcing approach with moving-grid process under arbitrary
Lagrangian-Eulerian (ALE) framework to simulate moving boundary
problems with heat transfer effect. Feng et al. (2009) combined direct
numerical simulation (DNS) with IBM to simulate the natural convection in
particulate flows. Kim et al. (2008) studied the natural convection induced by
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a temperature difference between a cold outer square enclosure and a hot inner
circular cylinder at different vertical locations using IBM, in which the heat
source/sink is introduced. Later, Lee et al. (2010) extended the work of Kim et
al. (2008) by considering various locations of inner cylinder along horizontal
and diagonal directions. In all the above works, the common feature is that a
heat source term is introduced in the energy equation. However, like the
conventional IBM, the heat source term is treated explicitly and pre-calculated.
In most of the works, there is no mechanism to enforce the boundary condition
for temperature. As a consequence, the boundary condition for temperature is
not accurately satisfied, which would affect the accuracy of numerical results.

Besides, almost all the efforts were made for heat transfer problems with
Dirichlet boundary condition for temperature, i.e., the temperature on the
immersed boundary is specified (without special illustration, the Dirichlet and

Neumann boundary conditions mentioned below specially refer to those on
immersed boundaries). The only notable one in applying IBM to thermal flows
with Neumann boundary condition was given by Zhang et al. (2008). In their
work, a layer of assistant points which are placed one-grid spacing away from
the immersed boundary along its outward normal direction is firstly defined.
Then the temperature at assistant points is calculated through interpolation
from the temperature at Eulerian points. After that, the normal derivative of
temperature in the Neumann condition is approximated by the first order
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one-sided finite difference scheme, from which the wall temperature can be
computed. With the calculated wall temperature, the algorithms used in the
work of Zhang & Zheng (2007) for fluid flows are applied to correct the
surrounding temperature field at Eulerian points. From the solution process of
Zhang et al. (2008), the Neumann condition is discretized to give the wall
temperature. Then the algorithms of IBM with Dirichlet condition are applied.
So in their method, the immersed boundary solver is employed to deal with the
Dirichlet condition. Technically, their method is not a real immersed boundary
solver for treating the Neumann condition. To the best of our knowledge, there
is no available work on the application of traditional IBM (Peskin’s original
IBM) for thermal flow problems with Neumann (given heat flux) boundary
condition for temperature in reported literatures. Compared to the Dirichlet
temperature condition case, heat transfer problem with Neumann-type
temperature condition presents an even a bigger challenge.

1.4 Objective of this thesis
The above literature review shows that remarkable works have been done on
the development of the immersed boundary method. However, there are still

rooms for the improvement of methodology and these stimulate the studies in
the present thesis. The objective of this thesis is to develop several novel
immersed boundary methods for fluid and thermal dynamics problems with
complex or moving boundaries, aiming to further refine the existing IBM. The
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Chapter 1 Introduction
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principal goals of the current study are as follows:
¾ The conventional IBM cannot guarantee an exact satisfaction of no-slip
condition on the immersed boundary, so the present thesis will firstly
extend the velocity correction-based IBM originally developed within the
framework of Lattice Boltzmann (LB) solver to the Navier-Stokes (NS)
solver-based version where the primitive variable formulation is solved.
¾ Although the stream function-vorticity formulation is more efficient for
two-dimensional incompressible flow, the published immersed boundary
models are so complicated that high computational effort is needed. The
present thesis will develop a computationally efficient stream
function-vorticity formulation-based immersed boundary method;
¾ The reported immersed boundary methods for thermal flows are not
capable of exactly satisfying the Dirichlet thermal condition on the
immersed boundary. In this thesis, a boundary condition-enforced IBM for
thermal flows with Dirichlet boundary condition will be presented.
Furthermore, two simple and convenient ways for the calculation of
Nusselt number in the framework of IBM are suggested;
¾ Very few work was reported by extending IBM to solve problems with
Neumann boundary condition in the framework of Peskin’s original IBM.
In the present thesis, an efficient heat flux correction-based immersed
boundary method will be proposed for heat transfer problems with
Neumann condition;