Development of immersed boundary methods for isothermal and thermal flows 4


Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
100



Chapter 4

A Boundary Condition-Enforced Immersed Boundary
Method for Heat Transfer Problems with Dirichlet
Condition
2


Heat transfer holds an important position in numerous engineering
applications. Their numerical simulations may present challenges when
complex geometries are involved. In this chapter, the IBM is extended to solve
heat transfer problems by modeling the heated immersed boundary as a set of
heat sources which is added to the energy equation as the source term. Based
on a temperature correction technique, a boundary condition-enforced
immersed boundary solver is developed for heat transfer problems with
Dirichlet-type boundary condition, in which the heat source/sink term is
considered as unknown and determined implicitly such that the energy
equation and the corresponding thermal boundary condition can be accurately
satisfied. Furthermore, the critical issue of how to effectively calculate the
average Nusselt number has not been properly addressed when IBM is applied
 
2
Parts of materials in this chapter have been published in
W.W. Ren, C. Shu, J. Wu, W. M. Yang, Computers & Fluids, 57 (2012) 40-51.


Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
101

to solve thermal problems. Another important contribution is that 2 efficient
ways to calculate the average Nusselt number in the framework of IBM are
suggested. They are based on temperature correction at Eulerian points and
heat source/sink at Lagrangian points, which completely avoid evaluation of
temperature gradients at the boundary points. The present method and
proposed ways to calculate the average Nusselt number are validated by their
applications to simulate forced convection over a stationary isothermal
circular cylinder and natural convection in a concentric annulus between a
square outer cylinder and a circular inner cylinder. The obtained numerical
results are very accurate and stable, and agree well with available data in the
literature.

4.1 Methodology
4.1.1 Governing equations
Assume that a cold fluid is flowing over a heated immersed body
Γ
inside
the two-dimensional domain
Ω
shown in Fig. 2.1. By extending Peskin's
original idea, the IBM can be used for solving the thermal flow problem, in
which the heated immersed boundary
Γ
is modeled as a set of singular heat
sources/sinks

Q
Δ
at each boundary segment (represented by Lagrangian
points). The heat source/sink at the Lagrangian point is then distributed into
the surrounding fluid as a volumetric heat source/sink
q
through the Dirac
delta function interpolation. As a result, the heat and mass transfer in the

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
102

framework of IBM inside the domain
Ω
are governed by

2
(()) p
t
ρμ
∂
+
∇=−∇+∇+
∂
u
uu ufi
(4.1a)

2

(()) (1( ))pTT
t
ρμρβ
∞
∂
+
∇=−∇+∇+ − − +
∂
u
uu u
g
fi
(4.1b)

0∇=ui
(4.2)


2
(())
p
T
cTkTq
t
ρ
∂
+∇ =∇+
∂
ui
(4.3)

with no-slip condition (2.3) and Dirichlet thermal condition (4.4) on
Γ


( (,),) ( (,),)
B
T stt T stt=XX
. (4.4)
Here Eqs. (4.1a), (4.2), (4.3) describe forced convection problems while Eqs.
(4.1b), (4.2), (4.3) describe natural convection problems. In Eq. (4.1b), the
Boussinesq approximation has been used.
T
denotes the fluid temperature.
β
is the thermal expansion coefficient due to temperature difference.
g
is
the gravitational acceleration directed downward.
p
c
is the specific heat at
constant pressure and
k
is the thermal conductivity at the reference
temperature
∞
= TT
. The heat source/sink term
q
in the energy equation

(4.3) is the heat density transferred to the fluid from the immersed boundary,
which, as mentioned, can be written as

( , ) ( ( , ), ) ( ( , ))qt Q stt stds
δ
Γ
=Δ −
∫
xXxX
(4.5)
Here,
((,),)QsttΔ X
is the boundary heat source/sink.

The solution procedure for velocity field
u
has been well described in

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
103

Chapters 2 and 3 (for two-dimensional problems, Eqs. (4.1) – (4.2) can be
transformed into the stream function-vorticity formulation if desired). In the
following, we exclusively focus on the calculation of temperature field and a
detailed description on how to derive an accurate temperature field will be
elaborated.

4.1.2 Temperature correction procedure
The boundary condition-enforced IBM for thermal flows developed in this

chapter aims at suggesting an IBM solver which can accurately satisfy both
the energy equation (4.3) and the Dirichlet temperature condition (4.4). By
using the fractional step approach, it is found that the solution of equation (4.3)
can be obtained by the Predictor-Corrector steps:

*
2
(())
n
p
TT
cTkT
t
ρ
−
+∇ =∇
Δ
ui
(4.6)

1*n
p
TT
c
q
t
ρ
+
−
=

Δ
(4.7)
The Predictor step (4.6) solves the energy equation without considering the
heat source/sink term q. The resultant solution is noted as predicted
temperature
*
(,)Ttx
, which can be calculated by solving the following
discretized equation

*
1* 1 2*2
(( ) ( ) ) ( )
22
n
p
nnn n
p
c
TT k
cTTTT
t
ρ
ρ
++
−
=− ∇ + ∇ + ∇ +∇
Δ
uuii

(4.8)

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
104

where the second-order Crank-Nicolson scheme is applied to the terms
()T∇ui
and
2
kT
∇
.

To obtain the desired temperature
1n
T
+
(x)
, the corrector step (4.7) is
implemented. It can be observed clearly from Eq. (4.7) that adding a heat
source/sink is equivalent to making a temperature correction

()
()
p
q
Tt
c
ρ

Δ= Δ
x
x
, (4.9)
which is responsible for correcting the predicted temperature field to the desire
one through

1*
() () ()
n
TTT
+
=+Δxx x
(4.10)
The heat source/sink
q
at the Eulerian grid point, as shown in Eq. (4.5), is
distributed from the boundary heat source/sink
Q
Δ
through Dirac delta
function interpolation and can be expressed in the discrete form as (if the same
spatial discretization scheme as in Section 2.4.2 is implemented)

( ) ( ) ( ) ( 1,2, , ; 1,2, , )
jihjii
i
qQDsiMjN=Δ − Δ = =
∑
xXxX 

(4.11)
Substituting Eq. (4.11) into Eq. (4.9) leads to

()
( ) ( ) ( 1,2, , ; 1,2, , )
i
jhjii
i
p
Qt
TDsiMjN
c
ρ
ΔΔ
Δ= −Δ= =
∑
X
xxX


(4.12)
Note that the unknowns in Eq. (4.12) are the boundary heat
sources/sinks
()
i
Q
Δ
X
. Once they are determined, the temperature correction


Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
105

and then corrected temperature can be calculated from Eqs. (4.12) and (4.10)
respectively. So, the key issue that requires to be addressed in the proposed
IBM model is the calculation of boundary heat sources.

In an attempt to satisfy the temperature boundary condition (4.4), it should be
ensured that the temperature at the boundary point interpolated from the
corrected temperature field by the delta function interpolation is equal to the
specified temperature
1
()
n
Bi
T
+
X
, that is,

11 2
() ()( ) (1,,; 1,,)
nn
Bi jhji
j
TTDhiMjN
++
=−==

∑
XxxX 

(4.13)
Substituting Eqs. (4.10) and (4.12) into Eq. (4.13) gives

1* 2
2
() ()( )
()
( ) ( )
n
Bi jhji
j
k
hj k khj i
jk
p
TTDh
Qt
DsDh
c
ρ
+
=−
ΔΔ
+−Δ−
∑
∑∑
XxxX

X
xX xX
(4.14)

11(,,; ,,)iMjN==

Equation (4.14) can be put in the following matrix form

[][][]
TT T
=CQ D
(4.15)
where
[]
11 12 1 11 12 1
12
21 22 2 21 22 2
2
12
12 1 2
12
NM
hh h h h hM
NM
hh h h h hM
T
p
MM MNN N NM
hh h h h hM
DD D DsDs Ds

DD D DsDs Ds
t
h
c
DD D DsDs Ds
ρ
⎛⎞⎛ ⎞
ΔΔ Δ
⎜⎟⎜ ⎟
ΔΔ Δ
Δ
⎜⎟⎜ ⎟
=
⎜⎟⎜ ⎟
⎜⎟⎜ ⎟
⎜⎟⎜ ⎟
ΔΔ Δ
⎝⎠⎝ ⎠
C


  


(4.16)

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
106


[]
1
11 12 1 *
,1
1
1
21 22 2 *
,2
2
2
1
12 *
,
n
N
B
hh h
n
N
B
hh h
T
n
MM MN
BM
hh hN
T
DD DT
T

DD DT
h
T
DD D T
+
+
+
⎛⎞
⎛⎞⎛⎞
⎜⎟
⎜⎟⎜⎟
⎜⎟
⎜⎟⎜⎟
=−
⎜⎟
⎜⎟⎜⎟
⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟
⎝⎠⎝⎠
⎝⎠
D





(4.17)

[]
1
2
T
M
Q
Q
Q
Δ
⎛⎞
⎜⎟
Δ
⎜⎟
=
⎜⎟
⎜⎟
Δ
⎝⎠
Q

(4.18)
1
,
(1,,)
n
Bi
Ti M
+
= 
,

*
(1,,)
j
Tj N= 
and
1
i
Qi M
Δ
=(,,)
represent
1
()
n
Bi
T
+
X
,
*
()
j
T x
and
i
QΔ (X )
, respectively.

As shown above, the nature of temperature correction procedure is that the
heat source/sink is determined in such a way that the temperature

( (,),)TsttX

at the boundary point interpolated from the corrected temperature field
),( tT x

equals to the specified boundary temperature

B
Tstt(X( , ), )
. In other
words, the physical boundary condition is enforced. After the equation system
(4.15) is solved,
i
QΔ
at all Lagrangian points are obtained. They are then
substituted into Eq. (4.12) to obtain the temperature correction
j
TΔ
, which is
further substituted into Eq. (4.10) to get the corrected temperature
),,1( NjT
j
=
.

4.1.3 Computational sequence
The basic solution procedure of the proposed method can be outlined below:
1)
Calculate the velocity field
u

using the method suggested in Chapter 2
or 3;

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
107

2) Solve Eq. (4.8) to get the predicted temperature
*
T .
3)
Solve equation system (4.15) to obtain the boundary heat source
i
QΔ

(
Ni ,,1=
) at Lagrangian points and then use Eq. (4.12) to get the
temperature correction
TΔ
.
4)
Correct the fluid temperature at Eulerian points using Eq. (4.10). Until
now, both the velocity field and temperature field have been updated to
time level
1n +
.
5)
Use the corrected velocity and temperature as the initial conditions, and
repeat steps 1 to 5 for the computation of next time level. The process

continues until a converged solution is achieved (steady case) or the given
time is reached (unsteady case).

4.2 Evaluation of Average Nusselt Number
Nusselt number is a key parameter in the heat transfer problem. The local
Nusselt number
Nu
is defined as

()
()
(,)
(,)
cref
hstL
Nu s t
k
=
X
X
(4.19)
where
()
(,)
c
hstX
is the local convective heat transfer coefficient,
ref
L
is the

reference length. According to Newton's cooling law and Fourier's law, the
heat conducted away from the wall by the fluid is equal to the heat convection
from the wall, that is

()
(,) ( (,))( )
cB
T
ksthstTT
n
∞
∂
−=−
∂
XX
(4.20)

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
108

Substituting Eq. (4.20) into Eq. (4.19) gives

() ()
1
ref
ref
BB
T
kL

L
T
n
Nu s t s t
TT k TTn
∞∞
∂
−
∂
∂
==−
−−∂
X( , ) X( , )
(4.21)
The surface-averaged Nusselt number
Nu
is an important parameter to
examine the rate of heat transfer from the heated surface, which is defined as

() ()
11
(,) (,)
ref
total total B
L
T
Nu Nu s t ds s t ds
LLTTn
∞
ΓΓ

∂
=⋅=−
−∂
∫∫
XX
(4.22)
where
tota l
L
is the total length of the immersed boundary

total
L
ds
Γ
=
∫
(4.23)
As shown in Eqs. (4.21) and (4.22), the evaluation of local and average
Nusselt numbers involves the calculation of the temperature gradient at the
boundary point. This chapter proposes two simple and efficient ways to
calculate the average Nusselt number directly from the heat source at Eulerian
points or the heat source at Lagrangian points, which completely avoid the
evaluation of the temperature gradient at Lagrangian points. Numerical
experiments show that the proposed ways can give very accurate and
consistent results for the calculation of average Nusselt number. In the
following, we will show the 2 ways in detail. To simplify the illustration, the
two-dimensional case is presented here. It should be noted that the suggested
methods can be naturally extended to three-dimensional problems.


Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
109

4.2.1 Method 1: Direct evaluation of average Nusselt number from
temperature correction at Eulerian points
As shown in Eq. (4.5), the heat source at an Eulerian point is from the heat
source at a few Lagrangian points. If we consider all the Eulerian points which
receive heat from the immersed boundary, then from the energy conservation
law, we have

() ( )
(,)qd Q stds
ΩΓ
=Δ
∫∫
xx X
(4.24)
in which the LHS (left hand side) is the volume integral in the whole fluid
domain, and the RHS (right hand side) is the surface integral along the
cylinder wall. According to Fourier's law,
(
)
QstΔ X( , )
can be written as

() ()
(,) (,)

T
Qst k st
n
∂
Δ=−
∂
XX
(4.25)
Substituting Eq. (4.9) and Eq. (4.25) into Eq. (4.24) gives

()
p
TT
cd k stds
tn
Γ
ρ
Ω
Δ
∂
=−
Δ∂
∫∫
xX(,)
(4.26)
The RHS of Eq. (4.26) can also be written in terms of local Nusselt number
()
(,)Nu s tX
as
()

(
)
B total
B
ref ref
Nu T T kL
kNu st T T
T
kstds ds
nLL
ΓΓ
∞
∞
−
⋅−
∂
−= =
∂
∫∫
()
X( , ) ( )
X( , )
,
(4.27)
where
B
T
is average temperature on the immersed boundary.
So, Eq. (4.26) can be simplified to

B
total p
ref
Nu T T
T
kL c d
Lt
ρ
∞
Ω
−
Δ
=
Δ
∫
()
x
(4.28)

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
110

With approximation of volume integral by the mean theorem, Eq. (4.28) can
be finally written as

(1,,)
()
j
pjj

j
ref
total B
T
cxy
t
Nu L j N
kL T T
ρ
∞
Δ
ΔΔ
Δ
==
−
∑

(4.29)
Note that variables on the RHS of Eq. (4.29) are dimensional. The actual form
of Eq. (4.29) for non-dimensional variables depends on the reference variables
for the problem considered. In the following numerical examples, the specific
form of Eq. (4.29) will be given.

4.2.2 Method 2: Direct evaluation of average Nusselt number from heat
flux at Lagrangian points

The heat source at Lagrangian points obtained from equation system (4.15)
can be directly used to compute the average Nusselt number, that is,

()

1
ref
total B
L
Nu Q s t ds
LkTT
Γ
∞
=Δ
−
∫
X( , )
()
(4.30)
The discrete form of Eq. (48) is

(1,,)
()
ii
i
ref
total B
Qs
Nu L i M
kL T T
∞
ΔΔ
==
−
∑


(4.31)
The above equation is also based on dimensional variables. Its specific form
for non-dimensional variables will be provided in the numerical examples.

4.3 Numerical Examples
The proposed boundary condition-enforced IBM, using the temperature

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
111

correction technique, is first numerically analyzed in this section through a
model problem of heat conduction. Its capability and efficiency are then
validated by its application to simulate both a forced convection (forced
convection over a stationary isothermal circular cylinder) and a natural
convection (natural convection in a concentric annulus between an outer
square cylinder and an inner circular cylinder) problem.

4.3.1 Numerical analysis of spatial accuracy
In this subsection, a model problem is used to investigate the spatial accuracy
of the proposed temperature correction-based IBM solver. The problem is the
heat conduction inside a concentric annulus between an outer square cylinder
(with size
11,11
x
y−≤ ≤ −≤ ≤
) and an inner circular cylinder (with radius
0.5r =
) as shown in Fig. 4.1. The governing equation with the use of IBM is

written as

2
T
Tq
t
∂
=∇ +
∂
(4.32)
The temperature boundary conditions are specified as:
1
in
T
=
on the inner
boundary and
(
)
22
1ln2
out
Txy=+ +
on the outer boundary, with the initial
condition
0T = everywhere in the internal region. The problem is solved
using five levels of uniform meshes with decreasing mesh spacing
1
16
h =

,
1
32
,
1
64
,
1
128
and
1
256
, respectively. The convergence criterion is set as

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
112

18
110
nn
TT
+−
∞
−≤×. The spatial accuracy is measured by
1
L
norm of the
relative error of temperature distribution, which is defined in the same way as
in Griffith & Peskin (2005); Mori & Peskin (2008). Fig. 4.2 shows the relative

1
L
error of the numerical solution with respect to the mesh spacing. The slope
of the line, which is 2, shows the second order of spatial accuracy.

4.3.2 Forced convection over a stationary isothermal circular cylinder
Forced convection over a stationary circular cylinder has been extensively
studied and used as a benchmark to examine the capability of new numerical
methods. Many experimental (Roshko 1961; Grove & Shair 1964; Kuehn &
Goldstein 1976; Sparrow et al. 2004) and numerical results (Dennis et al. 1968;
Dennis & Chang 1970; Lange et al. 1998; Mittal 1999; Soares et al. 2005;
Dhiman et al. 2006; Bharti et al. 2007) are available. This is a one-way
interaction problem and only the velocity field can influence the temperature
field. The fluid flow and heat transfer are characterized by the Reynolds
number
μ
ρ
DU
∞
=Re
and Prandtl number
k
c
p
μ
=Pr
. The same
computational domain as in Fig. 2.2 and the same domain discretization as in
Section 2.6.1 is taken for numerical simulation. Meanwhile, the same velocity
boundary conditions as those in Section 2.6.1 are applied. Generally, the

temperature is normalized by

B
TT
T
TT
∞
∞
−
′
=
−
(4.33)

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
113

where
B
T
is the uniform temperature on the cylinder surface and
∞
T
is the
free stream temperature. Then the boundary conditions for temperature are set
as:

0
T

y
′
∂
=
∂
on the top and bottom boundaries

0T
′
=
at the inflow boundary

0
T
x
′
∂
=
∂
at the outflow boundary

1
B
T
′
=
on the cylinder surface
In this study, numerical simulations are conducted for several low Reynolds
numbers of
Re 10, 20, 40=

and a fixed Prandtl number of
Pr 0.7=
, with
the convergence criteria
18
110
nn
TT
+
−
∞
−≤× . Since fluid flow
characteristics like the streamlines, drag coefficient
D
C
, recirculation length
/
w
LD
behind the cylinder have already been reported in Section 2.6.1, here
only heat flow characteristics such as isotherm patterns and average Nusselt
number on the cylinder surface are presented and compared with published
results (Dennis et al. 1968; Lange et al. 1998; Sparrow et al. 2004; Soares et a.
2005; Bharti et al. 2007).

Fig. 4.3 shows isotherms in the vicinity of the cylinder at
Re 20 40
=
,
. As can

be seen, the temperature contours (isotherms) at different Reynolds number
are topologically the same. They cluster heavily in the front surface of the
cylinder, indicating a large temperature gradient there. This implies that the

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
114

heat transfer rate at the front surface of the cylinder is much larger than other
regions. The clustering of isotherms is enhanced with increase of Reynolds
number
Re
.

As pointed out in Section 4.2, the average Nusselt number
Nu
is an
important parameter for heat transfer problems. Two ways to evaluate the
average Nusselt number are shown in Section 4.2. Their specific forms for this
particular problem are

RePr
1
j
jj
j
T
Nu x y j N
t
π

Δ
=ΔΔ=
Δ
∑
(,,) (4.34)
and

(1,,)
ii
i
Qs
Nu i M
π
ΔΔ
==
∑

(4.35)
respectively. In the following, we will take the case of
10Re =
as an
example to illustrate the performance of these two ways using different mesh
sizes. The obtained numerical results using the two ways are shown in Table
4.1. It is obvious from Table 4.1 that Method 1 and Method 2 provide almost
the same results. They converge well to stable solutions when the mesh size is
refined. Table 4.2 lists the comparison of computed average Nusselt numbers
for Re = 10, 20 and 40. Note that since the two methods give almost the same
results, only the results obtained using Method 2 are displayed. Table 4.2
shows that the present results basically agree well with reference data in the
literature. As expected, the average Nusselt number

Nu
on the cylinder

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
115

surface increases with Reynolds number
Re
.

4.3.3 Natural convection in a concentric annulus between a square outer
cylinder and a circular inner cylinder
To further validate the capability of the proposed method, a natural convection
in a concentric annulus between a square outer cylinder and a heated circular
inner cylinder is simulated. Unlike the forced convection problem discussed in
Section 4.3.2, the velocity and temperature fields are strongly coupled for the
problem considered in this section. The buoyancy force is the driving force for
the flow, and the Boussinesq approximation is used. The schematic diagram
for the computational configuration is shown in Fig. 4.4. For this problem,
heat is generated uniformly within the inner circular cylinder with radius
r
,
which is placed concentrically within the cold square cylinder with side length
L
. The no-slip and isothermal conditions are imposed on both cylinder walls.
The flow behaviour of this problem depends on the Prandtl number
P
r
,

Rayleigh number
23
()
p in out
cgLTT
Ra
k
ρβ
μ
−
=
, and aspect ratio
/(2 )
A
RL r=
,
where
in
T
is the temperature on the inner cylinder surface and
out
T
is the
temperature on the outer cylinder surface,
g
is the amplitude of gravity
acceleration. The aspect ratio
AR can uniquely determine the configuration of
the problem. In the present study, numerical investigations are conducted for
three different aspect ratios (

/2 5.0,AR L r
=
=
2.5, 1.67) and Rayleigh

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
116

numbers of
4
110Ra =×
,
5
110×
and
6
110×
with Prandtl number fixed at
7.0Pr =
. The initial conditions are set to be zero for
u
and
out
T
for
T
in
the entire domain. The side length of the outer cylinder,
L, is taken as the

reference length, and the temperature is normalized by

out
in out
TT
T
TT
−
′
=
−
. (4.36)
Uniform meshes are used for all the simulated cases, with mesh sizes of
/128hL= for
4
110Ra =×
and / 256hL
=
for
5
110Ra =×

and
6
110× ,
respectively. The convergence criteria are set as
16
110
nn
+

−
∞
−<×uu and
18
110
nn
TT
+−
∞
−<×.

Numerical results in terms of streamlines, isotherms and average Nusselt
number
Nu
are presented. For the concentric annulus, the flow and
thermal fields are symmetric about the vertical central line through the center
of the annulus. This can be observed clearly in Figs. 4.5-4.7, which show the
streamlines and isotherms for different cases. When
4
10=Ra
as shown in
Fig. 4.5, the heat transfer in the annulus is mainly dominated by conduction,
and the velocity field is too weak to affect the temperature distribution. The
isotherms in Fig. 4.5 show a series of concentric circular-like shapes around
the inner cylinder. However, due to the effect of buoyancy, the thermal
boundary layer on the bottom surface of the inner cylinder is slightly thinner
than that on its top surface. The circulations of flow show that as the aspect

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition

117

ratio
Ar
increases from
67.1
to
5
, the distance between the inner cylinder
and the walls of the outer cylinder increases. As a result, the two secondary
eddies embedded in the primary eddy at
1.67AR
=
and
2.5AR =
are
merged into one primary eddy at
5AR
=
on both sides of the annulus. When
Rayleigh number is increased to
5
10=Ra
as shown in Fig. 4.6, a plume
begins to appear on top of the inner cylinder. As a result, the thermal boundary
layer on the surface of the inner cylinder becomes thinner as compared to that
at
4
10=Ra
. Also, the centers of symmetric eddies move upward. For the

case of
1.67AR =
, the inner circular cylinder is large and the gap between the
inner and outer cylinders is small. Therefore, the convective flow induces the
formation of two additional symmetric vortices on the top of the annulus. For
the case of
2.5AR =
, the two secondary eddies embedded in the primary
eddy on both sides of the annulus at
4
10=Ra
are merged into a primary
vortex at
5
10=Ra
. As Rayleigh number increases further up to
6
10=Ra

(Fig. 4.7), due to increasing effect of convection, the core of the symmetric
primary vortices keeps moving upward. The plume formed on top of the inner
cylinder becomes stronger and drives the flow impinging on the top walls of
the outer cylinder, which leads to a thinner thermal boundary layer and denser
isotherm gradient on the surface of the inner cylinder and top wall of the
enclosure. As a consequence, the heat transfer in these regions is enhanced.
The size of the pair of symmetric vortices formed on the upper side of the
annulus for the case of
1.67AR
=
at

5
10=Ra
greatly increases while the

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
118

inner secondary eddies become squeezed and small. Also, small vortices
appear in the vicinity of bottom wall of the annulus for the case of
2.5AR =

and
5AR =
owing to separation of strong convection flow. In addition, the
size of the tiny vortices is slightly larger for
5AR
=
than that for
2.5AR =

due to larger gap between the inner and outer cylinder surfaces.

To further demonstrate the capability of Method 1 and Method 2, the average
Nusselt number is also computed for this problem and compared with
reference data in the literature. Here, the average Nusselt number
Nu
is
defined as
in in

hS
Nu
k
=
, where
in
S
is taken as half of the circumferential
length of the inner cylinder surface and
in
h
is the average heat transfer
coefficient of the inner cylinder surface. In this natural convection case, Eqs.
(4.29) and (4.31) can be simplified as

(1,,)
2( )
j
pjj
j
in out
T
cxy
t
Nu j N
kT T
ρ
Δ
ΔΔ
Δ

==
−
∑

(4.37)

(1,,)
2( )
ii
i
in out
Qs
Nu i M
kT T
ΔΔ
==
−
∑

(4.38)
Note that for this problem, the side length is the reference length,
in out
TT−
is
the reference temperature,
2
p
cL k
ρ
/

is the reference time. So,
Nu
can be
expressed in terms of the non-dimensional variables as

(1,,)
2
j
jj
j
T
xy
t
Nu j N
Δ
ΔΔ
Δ
==
∑

(4.39)

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
119


(1,,)
2
ii

i
Qs
Nu i M
ΔΔ
==
∑

(4.40)
The computed average Nusselt number
Nu
using the two methods are
compared with those of Shu & Zhu (2002) and Moukalled & Acharya (1996)
in Table 4.3. It can be seen from the table that both Method 1 and Method 2
give exactly the same values of calculated average Nusselt number, which
agree very well with the two sets of reference data. Table 4.3 also reveals that
the average Nusselt number
Nu
greatly depends on Rayleigh number
Ra

and aspect ratio
A
R
.
Nu
increases with the increase of
Ra
due to the
effect of buoyancy-induced convection, while it decreases with the increase of
A

R
due to the effect of widened annulus gap space.

4.4 Conclusions
In this chapter, a boundary condition-enforced immersed boundary method is
developed for heat transfer problems. The effect of thermal boundaries on the
flow and temperature fields is considered through the velocity correction and
temperature correction. In particular, a heat source term, which is distributed
from the boundary heat source via a discrete delta function, is introduced into
the energy equation. From the viewpoint of fractional step approach, this term
is equivalent to making a correction in the temperature field. The temperature
correction is evaluated implicitly in such a way that the temperature at the
immersed boundary interpolated from the corrected temperature field satisfies

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
120

the physical boundary condition. Another important contribution of this
chapter is that it presents 2 simple and efficient ways to compute the average
Nusselt number by directly using the temperature correction at Eulerian points
and the boundary heat source at Lagrangian points. Note that the temperature
correction at Eulerian points and the boundary heat source at Lagrangian
points are part of the numerical solution.

The present solver has proven to be of second order spatial accuracy through a
numerical analysis. The efficiency and capability of the present method and 2
ways to calculate average Nusselt number are validated by applying them to
simulate both forced convection and natural convection problems. Numerical
results showed good agreement with available data in the literature. It is

believed that the present method has a promising potential for solving heat and
mass transfer problems with Dirichlet-type boundary conditions.

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
121




Table 4.1 Comparison of average Nusselt number obtained
by the two proposed methods (Re=10)
Eulerian mesh size
560 400
×

840 600
×

1120 800×

No. of Lagrangian
points
40 60 60 90 80 100
Method 1 1.9163 1.9160 1.9154 1.9158 1.9150 1.9150
Method 2 1.9162 1.9160 1.9153 1.9158 1.9138 1.9150



Table 4.2 Comparison of average Nusselt numbers

References Re=10 Re=20 Re=40
Present 1.9150 2.5238 3.3519
Dennis et al. (1968) 1.8673 2.5216 3.4317
Lange et al. (1998) 1.8101 2.4087 3.2805
Soares et al. (2005) 1.8600 2.4300 3.2000
Sparrow et al. (2004) 1.6026 2.2051 3.0821
Bharti et al. (2007) 1.8623 2.4653 3.2825










Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
122





Table 4.3 Comparison of computed average Nusselt numbers
Case References
Ra

A

r

Present Shu &
Zhu (2002)
Moukalled &
Acharya (1996)
Method 1 Method 2
4
101×

1.67 5.303 5.303 5.40 5.826
2.5 3.161 3.161 3.24 3.331
5 2.051 2.051 2.08 2.071
5
101×

1.67 6.171 6.171 6.21 6.212
2.5 4.836 4.836 4.86 5.08
5 3.704 3.704 3.79 3.825
6
101×

1.67 11.857 11.857 12.00 11.62
2.5 8.546 8.546 8.90 9.374
5 5.944 5.944 6.11 6.107



Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition

123









Fig.4.1 Configuration for the model problem



Fig. 4.2
1
L
-norm of relative error of the temperature versus the mesh spacing
for the model problem


slope=1.92

Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems
with Dirichlet Condition
124


20() Rea
=

40() Reb
=

Fig. 4.3 Isotherms for flow over a heated stationary cylinder at
20 40Re ,=




Fig. 4.4 Schematic view of natural convection in a concentric annulus