An Immersed Boundary Method To Solve Flow And Heat Transfer Problems Involving A Moving Object

An immersed boundary method to solve flow and heat transfer
problems involving a moving object
Dedy Zulhidayat Noora
, Ming-Jyh Chern*,a
, Tzyy-Leng Horngb
a
Department of Mechanical Engineering, National Taiwan University of Science and Technology,
43 sec. 4 Keelung Road, Taipei 10607, Taiwan
b
Department of Applied Mathematics, Feng Chia University,
100 Wenhwa Road, Taichung 40724, Taiwan
Abstract
An immerse boundary method with both virtual force and heat source is developed
here to solve Navier-Stokes and the associated energy transport equations to study
some thermal flow problems caused by a moving rigid solid object within. The key
point of this novel numerical method is that the solid object, stationary or moving,
is first treated as fluid governed by Navier-Stokes equations for velocity and pressure,
and by energy transport equation for temperature in every time step. An additional
virtual force term is then compensated to the right hand side of momentum equations
at the solid object region to make it acting mechanically like a solid rigid body im-
mersed in fluid exactly. Likewise, an additional virtual heat source term is applied to
the right hand side of energy equation at the solid object region to maintain the solid
object at prescribed temperature all the time. The current method was validated by
some benchmark forced and natural convection problems such as a uniform flow past a
heated circular cylinder, and a heated circular cylinder inside a square enclosure. We
further demonstrated this method by studying a mixed convection problem involving a
heated circular cylinder moving inside a square enclosure. Our current method avoids
the otherwise requested dynamic grid generation in traditional method and shows great
efficiency in the computation of thermal and flow fields caused by fluid-structure inter-
∗
Corresponding Author.
E-mail : mjchern@mail.ntust.edu.tw
Tel: 886-2-27376496 Fax: 886-2-27376460
Preprint submitted to Elsevier January 21, 2010

action.
Key words: Immersed boundary, conjugate heat transfer, volume of solid,
fluid-structure interaction
2

Nomenclature
A dimensionless amplitude
d dimensionless displacement
D dimensionless diameter
f virtual force
F total force acting on solid body
Gr Grashof number, gβ(TH-TC)L3
/ν2
H total heat transfer over body surface
L dimensionless length
m time step level
n normal direction
Nus local Nusselt number
Nu total average Nusselt number
p dimensionless pressure
P dimensionless parameter
Pr Prandtl number, ν/α
r dimensionless gyration radius
R dimensionless radius
Ra Rayleigh number, GrPr
Re Reynolds number, usD/ν
St Strouhal number, fL/u
t dimensionless time
T temperature, K
u dimensionless velocity
W dimensionless area
x, y horizontal and vertical cartesian coordinate
xr dimensionless recirculation length
3

Greek symbols
̟ non-dimensional oscillation angular frequency, ωD/us
ω oscillation angular frequency, s−1
θ non-dimensional temperature
ν kinematic viscosity of fluid, m2
· s−1
α thermal diffusivity, m2
· s−1
β coefficient of thermal expansion, K−1
Subscripts
f fluid
s solid
4

1. Introduction
Interactions between fluid and structures are frequently encountered in many engineer-
ing applications such as heat exchanger, stirring reactors, ship hydrodynamics, off-shore
and harbor structures etc. In general, fluid-structure interaction problems and multi-
physics problems are often too complex or even impossible to solve analytically, so they
have to be analyzed by means of experiments or numerical simulations (CFD).
Computational fluid dynamics has been undergoing significant expansion as the re-
sult of rapid development of computing technology. In accordance with it, tremendous
efforts have been contributed to make new innovation in CFD area. It is well known
that computational fluid dynamics is an attractive tool to solve fluid-structure inter-
action problems due to its convenience for altering model parameters. CFD is now a
sophisticated and powerful analysis technique which encompass a wide range of indus-
trial and non-industrial application areas.
In terms of computational fluid dynamics, when the complex immersed object
moves, the problem poses some difficulties, i.e. how to treat a complex and moving
object as well as to get accurate and efficient computation. The common method to
simulate the flow with a complex boundary is the conventional body-fitted. Regard-
ing a moving object, the Arbitrary Lagrangian Eulerian (ALE) numerical approach is
a method to accommodate the complicated fluid-structure interface. In the Eulerian
methods using a static coordinate frame, the fluid flows through the static computa-
tional mesh. On the other hand, in the Lagrangian methods, the mesh moves with the
fluid. There is no mass flux among computational cells. Arbitrary Lagrangian-Eulerian
(ALE) methods introduced in [1] appear to be a reasonable compromise between La-
grangian and Eulerian approaches, allowing the solution of a large variety of fluid
problems. The ALE method consists of several Lagrangian computational time steps
followed by a mesh rezoning and a conservative quantities remapping. The mesh rezon-
ing step smooths the Lagrangian computational mesh and avoids its distortion. During
the remapping step the conservative quantities are conservatively remapped from the
5

old Lagrangian mesh to the new smooth one. After remapping the Lagrangian compu-
tation continues until the next rezone/remap steps which introduce the Eulerian flavor
into the method allowing mass flux between computational cells. The rezone/remap
steps keep the quality of the moving mesh good enough during the whole computation
and are performed either regularly after fixed amount of Lagrangian time steps or when
mesh quality deteriorates under some threshold. Many scholars have described ALE
strategies to optimize accuracy, robustness, or computational efficiency (see for exam-
ple [2-6]). Nevertheless, mesh updating or re-meshing processes can be computationally
expensive for the ALE algorithm.
In particular, the immersed boundary method has been getting popular in the last
few years since it was introduced by Peskin [7] due to its capability to handle sim-
ulations for moving complex boundary with lower computational cost and memory
requirements than the body-fitted method. In this method, a fixed Cartesian grid and
a Lagrangian grid are employed for fluid and immersed object, respectively. The inter-
action between the fluid and the immersed boundary is linked through the spreading of
the singular force from the Lagrangian grid to the Cartesian grid and the interpolation
of the velocity from the Cartesian grid to the Lagrangian grid using a discrete Dirac
delta function. Furthermore, some modifications and improvements of this method have
been proposed by other researchers [8-11]. This method can be categorized as contin-
uous forcing method wherein a forcing term is added to the continuous Navier-Stokes
equations before they are discretized.
Instead of using a delta function to distribute force from the Lagrangian grid to the
Cartesian grid, Yusouf [12] introduced a new immersed boundary method, namely the
direct-forcing method. This method uses a forcing term determined by the difference
between the interpolated velocities on the boundary points and the desired boundary
velocities. This method is also known as a discrete forcing method since the forcing is
either explicitly or implicitly applied to the discretized Navier-Stokes equations. The
idea of the direct-forcing method has been used and developed successfully in some
applications [13-16].
6

Fadlun et al. [14] developed combined immersed boundary finite difference methods
for three dimensional complex flow simulations. Some cases including the flow inside an
IC piston/cylinder assembly at high Reynolds number were simulated successfully. The
major issue of their work is the interpolation of the forcing over the grid that determines
the accuracy of the scheme. They implemented three different procedures i.e. no in-
terpolation or stepwise geometry, volume fraction weighting and velocity interpolation.
They calculated the effect of those procedures on the accuracy of the scheme and found
that the last procedure which has a second order accuracy gave the best results.
Pertinent to heat transfer problem, Paravento et al. [17] introduced an immersed
boundary method for complex flow and heat transfer. They solved both a hot and
insulated square body located in a box-domain open at the two faces perpendicular
to the main stream direction. However, this method can not be simply extended to
immersed boundaries not aligned with the grid.
Vega et al. [18] proposed a general scheme for the boundary conditions in convection
and conduction heat transfer using an immersed boundary method. The momentum
and energy forcing are imposed into non-dimensional momentum and energy equations,
respectively and employed the projection method to solve the governing equations in
nonstaggered-grid configuration. They used the most general linear boundary condi-
tion to developed the interpolation scheme to solve the energy equation. Some cases
regarding conduction, forced and natural convection have been simulated in order to
verify the proposed method.
Pan [19] introduced an immersed boundary method on unstructured Cartesian
meshes for incompressible flows with heat transfer. The body is identified by a volume-
of-body (VOB) function analogous to the volume-of-fluid (VOF) function. The body
interior is viewed as being occupied by the same incompressible fluid as outside with a
prescribed velocity and temperature field. Since the body velocity is assumed incom-
pressible, the pressure field inside the body obeys the same governing equation as the
pressure outside. This VOB approach can also be applied to the temperature equation
with a Dirichlet boundary condition.
7

In this present work, we propose an immersed boundary method to simulate fluid-
solid interaction with heat transfer by adopting the direct-forcing method. The energy
equations for fluids and solids are solved simultaneously by giving volume-of-solid frac-
tion, η, to differentiate the calculation of fluid and solid parts, subsequently leads to
conjugate heat transfer between them. η denotes fraction of solid within a cell where
η equal to 1 and 0 for solid and fluid cells, respectively, and be fractional on boundary
cells. In the case of a moving object, for making the scheme straightforward and robust,
we ignore the fractional value of η. The values of η will be either 1 or 0 only. As an
example, for the case of circular cylinder, we define η by following the function
η(x, y, t) =



1, (x − xc(t))2
+ (y − yc(t))2
≤ R2
0, (x − xc(t))2
+ (y − yc(t))2
> R2
(1)
where xc(t) and yc(t) are the x and y coordinate of the cylinder center at particular
time t, respectively, and R is the radius of the cylinder. Furthermore, the cells number
1 and 3 in Fig. 1 belong to fluids (η = 0), on the other hand, the cells number 2
and 4 belong to solids (η = 1). It is the reason that we chose the volume of solid
(VOS) method instead of using velocity interpolation method even thought the later
as reported by Fadlun et al. [14] is better in accuracy. The accuracy of the present
method can be enhanced by implementing fine grid arrangement near by the solid-fluid
boundary. Details of the numerical procedure are presented in the following section.
Finally we simulated some problems including moving body cases to demonstrate the
capability of the present method in handling fluid-solid interaction with heat transfer.
2. Numerical methods
2.1. The governing equations
The continuity, momentum and energy equations for an incompressible Newtonian fluid
are written using the following assumptions:
1. there is no viscous dissipation
2. the gravity acts in the vertical direction
8

3. fluid properties are constant and fluid density variations are neglected except in
the buoyancy term (the Boussinesq approximation)
4. radiation heat exchange is negligible.
We use the modified non-dimensional governing equations as follow,
∇ · u = 0 (2)
∂u
∂t
+ ∇ · (uu) = −∇p + P1∇2
u + f + P2θez (3)
∂θ
∂t
+ ∇ · (uθ) = P3∇2
θ + P4η∇2
θ. (4)
To deal with a conjugate heat transfer problem, we use η to differentiate the calculation
between solid and fluid parts. The parameters P1, P2, P3 and P4 are defined according
to the scaling of Eqs. (2)-(4) and depend on the problem under analysis as described
below.
Forced and mixed convection
We non-dimensionalize length with L, velocity with U, time with L/U and pressure with
ρU2
. The dimensionless temperature is defined as θ = (T − T0)/(Tw − T0) , where Tw is
the wall/body temperature and T0 is a reference temperature. Furthermore, Reynolds
number, Prandtl number and Grashof number are defined as Re = UL/ν, Pr = ν/α
and Gr = (gβL/U2
)(Tw − T0)Re2
, respectively, where ν, α, g and β are the kinematic
viscosity, the thermal diffusivity of fluid, the gravitational acceleration and the coeffi-
cient of thermal expansion, respectively. Therefore, we define P1 = 1/Re, P2 = Gr/Re2
,
P3 = 1/RePr and P4 = αs = α/LU where αs is the dimensionless thermal diffusivity of
a solid.
Natural or free convection
For natural or free convection, we non-dimensionalize length with L, velocity with α/L,
time with L2
/α and pressure with ρα2
/L2
. The non-dimensional temperature is defined
as θ = (T − T0)/(Tw − T0) and the Grashof number becomes Gr = (gβL3
/ν2
)(Tw − T0).
9

Thence we have P1 = Pr, P2 = RaPr, P3 = 1 and P4 = αs = α/LU, where Ra = GrPr
is the Rayleigh number.
The virtual force term, f, in Equation (3) represents the action of a solid upon a fluid.
It is defined as the rate of momentum changes of a solid body which is proportional the
difference between the solid velocity at the (m + 1)th
time step and the fluid velocity at
the mth
time step. This force exists on the solid body and zero elsewhere which reveals
the existence of a force to hold or drive a solid body when it is stationary or moving.
Furthermore, it can be simply written as
fm+1
= η
um+1
s − um
∆t
, (5)
where us is the prescribed velocity of a solid body. We use a staggered grid arrangement.
That is, the pressure and the temperature are located at the center of the cell while the
velocities are placed at the faces of the cell. Note that u in Equation (5) is the velocity
at the center of the cell which is interpolated from its faces.
2.2. Numerical procedures
We use three and two-step time-split schemes to advance the flow and thermal fields,
respectively. First the velocity and temperature are advanced from the mth
time level
to the first intermediate level ” ∗ ” by solving the advection-diffusion equations without
the pressure and the virtual force for the momentum Eq. (3) and without conduction of
solid part for the energy Eq. (4). Subsequently, this step can be stated in the following
forms
u∗
− um
∆t
= Sm
+ P2θ, (6)
θ∗
− θm
∆t
= Hm
, (7)
where S and H are the advection and diffusion terms of the momentum and the energy
equations which are discretized using the second-order upwind and central difference
schemes, respectively. We used the second-order Adam-Bashford method for the tem-
poral discretization of those equations in such away that term S and H can be rewritten
10

as
Sm
=
3
2
(−∇ · (uu) + P1∇2
u)m
−
1
2
(−∇ · (uu) + P1∇2
u)m−1
(8)
and
Hm
=
3
2
(−∇ · (uθ) + P3∇2
θ)m
−
1
2
(−∇ · (uθ) + P3∇2
θ)m−1
. (9)
This intermediate velocity field, in general, does not satisfy the divergence-free condition
(2). At the second step we advance the first intermediate velocity by including the
pressure term
u∗∗
− u∗
∆t
= −∇pm+1
. (10)
Applying the divergence on both sides, Eq. (10) becomes
∇ · u∗∗
− ∇ · u∗
∆t
= −∇2
pm+1
. (11)
Due to conservation of mass, we have
∇ · u∗∗
= 0. (12)
Then substituting Eq. (12) to (11) gives the Poisson equation
∇2
pm+1
=
1
∆t
∇ · u∗
. (13)
Once Eq. (13) is solved, we can advance the intermediate velocity in Eq. (10). Fur-
thermore, we update the velocity to the (m+1)th
time level by imposing the force term
as follows
um+1
− u∗∗
∆t
= fm+1
. (14)
Similarly, we get the new temperature at the (m + 1)th
time level by imposing the
calculation of conduction of solid body
θm+1
− θ∗∗
∆t
= P4η∇2
θ∗
. (15)
The thermal boundary conditions on a solid surface are θ = 1 and θ = 0 for hot and cold
surface, respectively [20]. Once the velocity and temperature fields are obtained, the
local Nusselt number on the solid surface is evaluated using the following expressions
Nus =
∂θ
∂n
(16)
11

Afterward, surface-average Nusselt number becomes
Nu =
1
W
Z W
0
Nusds (17)
where n is the normal direction with respect to the walls and W is the total surface
area of walls.
Instead of using the integration method over solid surface to calculate total heat
transfer over the body surface, especially for the case of a moving object, it is more
convenient to transform it into a volume integral over the body volume in such a way
that
H =
N
X
i=1
αsηi∇2
θi∆∀i (18)
where ∀ and N are volume of the ith
cell and total number of cells, respectively. Simi-
larly, the forces acting on the immersed body can be calculated as
F =
N
X
i=1
fi∆∀i. (19)
Finally, a whole numerical procedure for each time step of the proposed method is given
below:
1. Detect the location of boundary cells and determine η.
2. Calculate the first intermediate velocity and temperature (Eqs. 6 and 7).
3. Solve the Poisson equation (Eq. 13) and advance the intermediate velocity (Eq.
10).
4. Calculate the virtual force (Eq. 5) and update velocity using the calculated virtual
force (Eq. 14).
5. Update temperature by imposing calculated heat conduction on solid part (Eq.
15).
12

3. Validation and numerical results
3.1. Forced convection over a heated circular cylinder with an isothermal surface
We simulated forced convection over a heated circular cylinder placed in an unbounded
uniform flow. The geometric set up in the computational domain and the associate
physical boundary conditions are shown in Fig. 2. Direchlet boundary condition i.e.
uniform velocity profile is applied at inlet and Neumann boundary conditions are ap-
plied at lateral and outlet sides. The incoming fluids are cold (θ = 0) whereas the
cylinder is considered as a heated body with an isothermal surface (θ = 1). A non-
uniform grid (220 × 180) is adopted to discretize the computational domain, with a
uniform grid (50 × 50) is employed to cover the cylinder.
Simulations have been conducted for different Reynolds numbers based on the cylin-
der diameter and the free stream velocity. Streamlines, vorticity contours and isotherm
contours for simulation at Re = 40 and Re = 100 are shown in Figures 3 and 4, respec-
tively. It is well known that at low Re, the flow pattern remains symmetric with a pair
of stationary recirculating vortices behind the cylinder (Fig. 3). Increasing Re leads to
instability of flow structures, so a pair of symmetrical vortices behind cylinder breaks
down and the vortex starts to shed up and down alternatively (Fig. 4). This shed-
ding frequency can be revealed as a dimensionless frequency, namely Strouhal number.
The recirculation length (xr), drag coefficient (CD), Strouhal number (St) and Nusselt
number (Nu) are compared with some previous works and presented in Table 1. In
general, all results obtained by the established model show good agreements with those
previous studies.
Since the present method is developed for a conjugate heat transfer problem also, we
further consider another case of a flow over unbounded circular cylinder. The geometry
for this case is the same as shown in Fig. 2. The incoming fluids are hot and the cylin-
der is cold initially. The cylinder surface is gradually heated by hot fluids. Heat from
fluids conducts towards the cylinder core. Due to the impinging effect of the incoming
hot fluid, the thermal field in the cylinder upstream is higher than the downstream one.
13

We plot the isotherm contours at different time step in Fig 5. However, when time goes
to infinity the whole cylinder will be in thermal equilibrium with its surrounding.
3.2. Natural convection in a square enclosure with a heated circular cylinder
We performed simulation of natural convection in a square enclosure with a heated
circular cylinder placed at its center. The system consists of a square enclosure with
sides of length L whereas the diameter of the cylinder is D = 0.2L. The walls of the
square enclosure are kept at a constant low temperature of θC while the cylinder is
kept at constant high temperature of θH . We define L, θC and θH as the characteristic
length, the reference temperature and the wall temperature, respectively. The compu-
tational domain as well as the boundary conditions is shown in Fig. 6. A uniform grid
distribution of (201 × 201) along horizontal (x) and vertical (y) directions is employed.
The isotherm and streamline contours for simulations at Ra = 104
, Ra = 105
and Ra =
106
are presented in Fig. 7. For the case at Ra = 104
heat transfer inside the enclosure
is mainly dominated by conduction. At higher Ra of 105
, the plume starts to appear
on the top of the cylinder. The thermal gradient in the upper part of the enclosure is
much stronger than the lower one, in consequence, the dominant flow is found in the
upper half of the enclosure that indicates convection plays an important role in flow
and thermal field inside the enclosure. As Ra increases up to 106
, the heat transfer
in the enclosure is mainly governed convection. The plume strongly impinges on the
top of the enclosure to form a thinner thermal boundary layer and enhances the heat
transfer. The total surface-average Nusselt numbers of the cylinder for different Ra are
given in Table 2. The total surface-average Nusselt number of the cylinder is higher
as Ra increases due to the domination of convective heat transfer. To study the effect
of fractional values of η, we performed simulations for two type of grid i.e. A and B
as shown in Table 2. The terms A and B indicate that the calculations are performed
by ignoring and including fractional values of η, respectively. η is either 0 or 1 only
for A and 0 ≤ η ≤ 1 for B. In terms of our results, it is observed that the difference
of calculation Nu between A and B is not significant when simulations are preformed
14

using more grids. Furthermore, we choose uniform 201 × 201 grids for all simulations
in the following cases.
To further validate the proposed method, we also performed simulations for natu-
ral convection over a heated cylinder placed eccentrically inside a square enclosure.
The cylinder has a diameter D = 0.2L, with its center being located at (x, y) =
(−0.15L, −0.15L) as measured from the center of the enclosure. This case is common
and has been studied by some other scholars. The isotherm and streamline contours
at different Ra are presented in Fig. 8 whereas the comparison of the average Nusselt
number with some previous works is given in Table 3. The comparisons for both con-
centric (Table 2) and eccentric case (Table 3) have shown good agreements between the
present method and some previous studies.
3.3. Mixed convection in a square enclosure with a moving heated circular cylinder
To further demonstrate ability the present method in handling fluid-solid interaction
with a moving object, we performed simulation of mixed-convection in a square enclo-
sure with a moving heated circular cylinder placed at its center. The cylinder has a
diameter D = 0.2L, where L is the length of the square enclosure, and is oscillating hor-
izontally with amplitude the A = 0.5D and the non-dimensional oscillating frequency
̟ = 1 as shown in Fig. 9. The instantaneous displacement of the cylinder d from its
mean position is given by d = A sin(̟t). We simulated the problem till it reaches a
periodic condition and take the instantaneous streamline and isotherm contours during
a period as shown in Fig. 10. Note that fluid and thermodynamic properties in this
case are as same as the case of natural convection at Ra = 106
.
In mixed convection, both natural and forced convection participate in the heat
transfer process. The forced flow can be either in the same direction (assisting mixed
convection) or in the opposing direction (opposing mixed convection) to the flow caused
by buoyancy. The natural convection causes a vertical plume. Moving the cylinder up
and down plays both natural and forced convection in an interactive way. Furthermore,
we consider this problem as a part of our simulations. The problem has the same geom-
15

etry and boundary conditions with the case in Fig. 9. The hot cylinder moves up and
down to create assisting and opposing mixed convection inside the enclosure. The in-
stantaneous isotherm, streamline and vorticity contours during a period are presented
in Fig. 11. In the case of horizontally moving cylinder (Fig. 10), two main eddies
occupy the entire enclosure while convective cells are shedding at the downstream of
the moving cylinder. This condition is repeated periodically in the opposite sense at
each half of period. On the other hand, for the case of a vertically moving cylinder
(Fig. 11), twin co-rotating eddies are formed when the cylinder moves down during
the opposing mixed convection process. Hence, the periodic isotherm, streamline and
vorticity contours are not presented during a complete period.
Starting from the horizontal and vertical motion of the cylinder, we extended our
simulation by combining those motions i.e. the cylinder moving in a counter-clockwise
circle orbit with its center located at the center of cavity and radius of r. Such a move-
ment stirs the flow inside the cavity in a more complicated way and may further cause
more complicated patterns in natural and forced convection. The description of this
problem is given in Fig. 12. The isotherm, streamline and vorticity contours during a
period for r = R are given in Fig. 13. Together with the previous cases, we show the
time histories of Nu for those cases and presented them in Fig, 14. For the considered
amplitude (A) and oscillating frequency of cylinder motion (̟), it can be observed that
Nu for moving objects are lower than the case of stationary one (natural convection
at Ra = 106
) i.e. 6.108. Nevertheless, to make it clear, let us consider the following
analysis. When the cylinder moves in any direction, it sheds and left convective cells
behind. These cells grow up to spread heat to adjacent fluids. After one period, while
the convective cells keep growing up, the cylinder moves toward its starting position
and penetrates the growing hot region, the heat transfer form the hot cylinder to the
fluid become ineffective consequently. To further observe these cases, we performed the
simulations at different amplitudes (A) and oscillating frequencies (̟) of body motion.
We tabulated the calculation of Nu in Table 4. When the amplitude is very small e.g.
A = 0.005D, the cylinder acts as a vibrating object. For all the cases in Table 4 we
16

observe that the highest Nu is found at the smallest amplitude, A = 0.005D. On the
other hand, smaller oscillating frequencies offer higher Nu due to slower motion of the
cylinder which spends more time for convective cells spreading heat before the cylinder
comes back to this region. Note that all those cases are compared to natural convec-
tion at Ra = 106
for the same fluid and thermodynamic properties which the flow is
more dominated by natural convection than by force convection. Hence, the motion of
cylinder does not offer significant any effect to increase heat transfer between the hot
cylinder and the fluid.
4. Conclusions
In the present study, we developed an immersed boundary technique to solve fluid-
solid interaction with conjugate heat transfer problems. A virtual force based on the
volume of solid is added to the Navier-Stokes equations. The fraction of solid (η) inside
computational domain is imposed to energy equation also to distinguish the calculation
of fluid and solid and accommodate conjugate heat transfer problem between fluid and
solid. Some benchmark cases including the moving heat transfer problem are simulated
to validate the proposed method in solving fluid-solid interaction with heat transfer
problems. As a result, the proposed immersed boundary method can be easily and
successfully applied to simulate heat transfer cases with a moving object as well as
conjugate heat transfer problem.
17

Acknowledgements
This work is supported in part by the National Science Council Grant 96-2115-M-035-
001, Taipei, Taiwan, by National Center for Theoretical Sciences (NCTS), Hsinchu,
Taiwan, and by Taida Institute for Mathematical Science (TIMS), Taipei, Taiwan.
Authors would like to thank university computing center for providing the computing
facility in the university cluster.
18

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for viscous incompressible flows with complex immersed boundaries, Journal of
Computational Physics 156 (1999), 209-240.
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of Computational Physics 161 (2000), 35-60.
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complex geometry, Journal of Computational Physics 192 (2003), 593-623.
[16] N. Zhang, Z.C. Zheng, An improved direct-forcing immersed-boundary method
for finite difference applications, Journal of Computational Physics 221 (2007),
250-268.
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complex flow and heat transfer, Flow Turbulence Combust. 80 (2007), 187-206.
[18] A.P. Vega, J.R. Pacheco, T. Rodic, A general scheme for the boundary conditions
in convective and diffusive heat transfer with immersed boundary methods, J. Heat
Transfer 129 (2007), 1506-1516.
20

[19] D. Pan, An immersed boundary method on unstructured Cartesian meshes for in-
compressible flows with heat transfer, Numerical Heat Transfer: Part B 49 (2006),
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heat transfer from an isothermal and isoflux sphere in the steady symmetric flow
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[21] J. Kim, D. Kim, H. Choi, An immersed-boundary finite-volume method for simula-
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[23] A. Diaz and S. Majumdar, Numerical computation of flow around a circular cylin-
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[26] B.S. Kim, D.S. Lee, M.Y. Ha and H.S. Yoon, A numerical study of natural con-
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[27] H. Sadat and S. Couturier, Performance and accuracy of a meshless method for
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21

Table 1: The comparison of average drag coefficients, recirculation length and Strouhal numbers at Re
= 40 and 100
Re = 40 Re = 100
CD xr Nu CD CL St Nu
Ecker and soehngen [22] - - 3.48 - - - -
Ye et al. [13] 1.52 2.27 - - - - -
Vega et al. [18] 1.53 2.28 3.62 - - - -
Lai and Peskin [0] - - - 1.4473 ±0.3229 0.165 -
Kim and Choi [2] 1.51 - - 1.33 ±0.32 0.165 -
Su et al. [11] 1.63 - - 1.4 - 0.168 -
Tseng and Ferziger [15] 1.53 2.21 - 1.42 ±0.29 0.164 -
Dias and Majumdar [23] 1.54 2.69 - 1.395 ±0.283 0.171 -
Pan [21] 1.51 2.18 3.23 1.32 ±0.32 0.16 5.02
Present study 1.567 2.219 3.32 1.4 ±0.322 0.167 5.08
22

Table 2: Nu around a circular cylinder placed concentrically inside an enclosure, D = 0.2L. The terms
A and B indicate that the calculations are performed by ignoring and including fractional values of η,
respectively. η will be only either 0 or 1 for A and 0 ≤ η ≤ 1 for B
Ra = 104
Ra = 105
Ra = 106
Moukalled and Archarya [24] 2.071 3.825 6.107
Shu et al. [25] 2.082 3.786 6.106
101×101 A 2.100 3.767 5.837
101×101 B 2.065 3.550 5.245
151×151 A 2.109 3.839 6.077
151×151 B 2.107 3.835 6.070
201×201 A 2.079 3.813 6.108
201×201 B 2.078 3.812 6.106
23

Table 3: Nu around a circular cylinder placed eccentrically inside an enclosure, D = 0.2L
Ra = 104
Ra = 105
Ra = 106
Sadat and Couturier [27] 4.699 7.430 12.421
Pan [19] 4.686 7.454 12.540
Vega et al. [18] 4.750 7.519 12.531
Present study 4.712 7.481 12.523
24

Table 4: Nu of a moving circular cylinder at different amplitude and oscillating frequency, D = 0.2L
motion ̟ A = 0.005D A = 0.05D A = 0.5D
horizontal 0.1 6.117 6.111 6.062
1.0 6.114 6.083 5.628
vertical 0.1 6.130 6.125 6.110
1.0 6.127 6.103 5.912
c.o.m 0.1 6.122 6.120 6.079
1.0 6.118 6.085 5.639
25

1
2
3 4
R
Solid
Fluid
Figure 1: Computational cells around solid-fluid boundary
26

Figure 2: Computational domain and boundary conditions
27

(a)
(b)
(c) 1.0731
1.0016
0.9300
0.8585
0.7870
0.7154
0.6439
0.5723
0.5008
0.4292
0.3577
0.2862
0.2146
0.1431
0.0715
0.0198
0.0048
Figure 3: (a) Streamlines, (b) vorticity contours and (c) isotherm contours at Re = 40
28

(a)
(b)
(c) 1.0787
0.9198
0.7608
0.6018
0.4837
0.4231
0.3634
0.3255
0.2996
0.2674
0.2246
0.1851
0.1597
0.1400
0.1296
0.1063
0.0884
0.0675
0.0455
Figure 4: (a) Streamlines, (b) vorticity contours and (c) isotherm contours at Re = 100
29

0.9928
0.9834
0.9576
0.9333
0.8666
0.7999
0.6666
0.5333
0.4000
0.2666
0.1555
0.0667
t=15
0.9937
0.9865
0.9823
0.9713
0.9570
0.8489
0.7183
0.5877
0.4571
0.3265
0.1959
0.0653
t=45
Figure 5: Temporal isotherm contours at Re = 100
30

Figure 6: Computational domain and the coordinate system along with boundary conditions
31

Ra = 104
Ra = 105
Ra = 106
0.937501
0.875001
0.812501
0.750001
0.687501
0.625001
0.562501
0.500000
0.437500
0.375000
0.312500
0.250000
0.187500
0.125000
0.062500
Figure 7: Isotherm (upper) and streamline (lower) contours of natural convection in a square enclosure
with a circular cylinder, R = 0.2L
32

Ra = 10
4
Ra = 10
5
Ra = 10
6
0.9375
0.8750
0.8125
0.7500
0.6875
0.6250
0.5625
0.5000
0.4375
0.3750
0.3125
0.2500
0.1875
0.1250
0.0625
Figure 8: Isotherm (upper) and streamline (lower) contours of natural convection in a square enclosure
with a circular cylinder, R = 0.1L
33

Figure 9: Computational domain and the coordinate system along with boundary conditions for mixed-
convection in a square enclosure with a moving heated circular cylinder
34

t = 0.25T t = 0.5T t = 0.75T t = T
0.9375
0.8750
0.8125
0.7500
0.6875
0.6250
0.5625
0.5000
0.4375
0.3750
0.3125
0.2500
0.1875
0.1250
0.0625
Figure 10: Temporal isotherm (upper) streamline (middle) vorticity (lower) contours during a period,
A = 0.5D
35

t = 0.25T t = 0.5T t = 0.75T t = T
0.9375
0.8750
0.8125
0.7500
0.6875
0.6250
0.5625
0.5000
0.4375
0.3750
0.3125
0.2500
0.1875
0.1250
0.0625
A = 0.5D
36

Figure 12: Computational domain and the coordinate system along with boundary conditions for
mixed-convection in a square enclosure with a heated circular cylinder moving in a counter-clockwise
orbit
37

t = 0.25T t = 0.5T t = 0.75T t = T
0.9375
0.8750
0.8125
0.7500
0.6875
0.6250
0.5625
0.5000
0.4375
0.3750
0.3125
0.2500
0.1875
0.1250
0.0625
A = 0.5D
38

t
Nu
50 55 60 65 70 75 80
4.5
5
5.5
6
6.5
horizontal motion
vertical motion
counter-clockwise orbital motion
Figure 14: Total average Nusselt number of cylinder for different kind of cylinder motion at ̟ = 1.
The length of the square enclosure is L while the diameter of the cylinder is D = 0.2L. A and r are
the amplitude and the radius of the cylinder motion, respectively.
39

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