Development of immersed boundary methods for isothermal and thermal flows 5


Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
128




Chapter 5

An Efficient Immersed Boundary Method for Thermal
Flow Problems with Heat Flux Boundary Condition
3


The exploration on generalizing Peskin’s original IBM to solve problems with
Neumann-type boundary condition is discussed in this chapter and we propose
a heat flux correction-based IBM for thermal flows with given heat flux
boundary condition. By using the fractional step procedure, when the
Neumann condition is not satisfied by the predicted temperature field, their
difference will contribute as a boundary heat source. Then following the
concept of conventional IBM, the boundary heat source will be distributed to
the surrounding Eulerian points as a volumetric heat source, which will be
used to correct the temperature field directly. In the whole process, the
Neumann condition is treated directly within the scope of immersed boundary
method. There is no need to define a layer of assistant points and to convert
the Neumann condition to the wall temperature. The present method is
 
3
Parts of materials have been published in
[1] C. Shu, W.W. Ren, W.M. Yang, Int J Numer Meth Heat Fluid Flow, 23 (2013) 124-142.

[2] W.W. Ren, C. Shu, W.M. Yang, Int. J. Heat Mass Transfer, 64 (2013) 694-705.


Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
129

validated by applying it to simulate forced convection over a stationary heated
circular cylinder and natural convection in a horizontal concentric annulus
between two circular cylinders. The obtained numerical results show that the
proposed IBM solver is suitable for addressing thermal flows subjected to heat
flux boundary condition accurately and efficiently.

5.1 Methodology
5.1.1 Governing equations
The same flow configuration
Ω
+
Γ
as the one in Fig. 2.1 is considered.
Assume that a thermal fluid is flowing inside it. Rather than specifying with
given temperatures as in Section 4.1.1, the boundary
Γ
herein is releasing
prescribed heat flux
B
Q
in its outward normal direction to the surrounding
fluid. Nevertheless, by representing the heated boundary
Γ

as a set of heat
sources at each boundary segment (represented by Lagrangian point), it shares
the same set of governing equations (4.1) – (4.3) and velocity boundary
condition (2.3) in the framework of IBM, while the temperature boundary
condition, under the current circumstance, is a Neumann-type one

((,)) ((,))
B
T
kstQst
n
∂
−=
∂
XX
on

Γ
, (5.1)
where
n
points to the outward normal direction of
Γ
. The heat source term
q
in the energy equation (4.3), as in the case of thermal flows subject to
specified temperature condition, is distributed from the boundary heat

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition

130

source
((),)QstΔ X
through (4.5).

The fluid flow field, i.e. velocity field, can be calculated following the
procedures suggested in Chapter 2 or 3. In the present discussion, we
exclusively focus on the temperature field and energy equation.

5.1.2 Heat Flux Correction Procedure
Predictor-corrector algorithm is a wonderful technique. It is extremely useful
when dealing with IBM and almost all the existing IBMs rely on the
Predictor-Corrector algorithm to fulfill their implementation. Following a
similar predictor-corrector step (4.6) – (4.7) as for the case of thermal flows
with specified temperature condition, the energy equation (4.3) together with
temperature boundary condition (5.1) can be successfully solved once the
volumetric heat source
q
is known. Therefore, the primary issue for the
whole solution process is the evaluation of boundary heat source
((),)QstΔ X

at each Lagrangian point, from which the volumetric heat source
q
could
become available through (4.5). However, the boundary heat source
determination is not an easy job. Several models have been examined during
our preliminary study, and finally, an efficient heat flux correction-based
solver, as will be elaborated in details in the following, is found to be effective

and accurate.


Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
131

Before we discuss the evaluation of
((),)Qst
Δ
X
, let us look at why q is
introduced in the energy equation. Note that Eq. (4.6) is the standard energy
equation for temperature without any heat source. If
*
T given in Eq. (4.6)
satisfies the heat flux condition (5.1), q should be taken as zero. From this
process, it is clear that the non-zero value of q is due to the fact that the heat
flux condition (5.1) is not satisfied by
*
T . Indeed, it is from their difference.
So, at first, we need to calculate
()
*
,
i
T
kt
n
∂

−
∂
X
at each Lagrangian point. To
do this, we can use discrete delta function interpolation (assuming the same
spatial discretization as in Section 2.4.2 is utilized) to provide

()
()
**
2
, , ( ) (1,,; 1,,)
ijhji
j
TT
ttDhiMjN
xx
∂∂
=−==
∂∂
∑
XxxX""
(5.2a)

()
()
**
2
, , ( ) (1,,; 1,,)
ijhji

j
TT
ttDhiMjN
yy
∂∂
=−==
∂∂
∑
XxxX""
(5.2b)
where
()
*
,
i
T
t
x
∂
∂
X
and
()
*
,
i
T
t
y
∂

∂
X
represent temperature derivatives with
respect to
x
and
y
at Lagrangian point
i
X
, while
()
t
x
T
j
,
*
x
∂
∂
and
()
t
y
T
j
,
*
x

∂
∂
are temperature derivatives with respect to
x
and
y
at Eulerian
point
j
x
. Note that the temperature derivatives at Eulerian points are obtained
by the second order central difference schemes. Finally, the temperature
derivative at the Lagrangian point is calculated by

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
132


() () ()
** *
,, ,
iixiiyi
TT T
ttntn
nx y
∂∂ ∂
=+
∂∂ ∂
XX X

(5.3)
Notice that in Eq. (5.3), the calculation of
()
*
,
i
T
kt
n
∂
−
∂
X
depends on the
normal direction
n
G
. The Neumann temperature condition (5.1) is also related
to
n
G
. For the application of IBM, the whole domain including interior and
exterior of the immersed object is used as the computational domain. Thus, at
a boundary point, there are two normal directions. One is to point to the flow
domain while the other is to direct into the inside of immersed object. The
boundary heat flux due to difference of
(
)
Bi
QtX,

and
()
*
,
i
T
kt
n
∂
−
∂
X
in the
two normal directions will both affect the temperature field at surrounding
Eulerian points. Therefore, when the difference of
()
Bi
QtX,
and
()
*
,
i
T
kt
n
∂
−
∂
X

are considered in the two normal directions,
()
,
i
QtΔ X
is
calculated as
() () () () ()
**
,2 , , 2 , ,
iBi i Bi i
TT
QtQtk t Qtk t
nn
⎡⎤
⎛⎞⎡ ⎤
∂∂
Δ= −− = +
⎢⎥
⎜⎟
⎢
⎥
∂∂
⎝⎠⎣ ⎦
⎣⎦
XX X XX

(5.4)
Note from Eq. (4.5) that the volumetric heat source
q

at the Eulerian grid
point is evaluated from the boundary heat source
Q
Δ
through Dirac delta
function interpolation, which can be expressed in the following discrete form

(,) (,) ( ) ( 1,, ; 1,,)
jihjii
i
qt QtD si Mj N=Δ − Δ = =
∑
xXxX""

(5.5)

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
133

Substituting Eq. (5.4) into Eq. (5.5) leads to

() ()
*
(,) 2 , , ( )
( 1, , ; 1, , )
j
Bi i hj ii
i
T

qt Q tk tD s
n
iMjN
⎛⎞
∂
=+ −Δ
⎜⎟
∂
⎝⎠
==
∑
xXXxX
""
(5.6)
With calculated q from Eq. (5.6), the temperature correction can be computed
from Eq. (4.9), and the corrected temperature field is obtained by Eq. (4.7).

It should be noted that although Zhang et al. (2008) has ever applied the
concept of immersed boundary to thermal flows with Neumann conditions,
they suggested to first define a layer of assistant points which are placed
one-grid spacing away from the immersed boundary along its outward normal
direction. With the help of these assistant points, the normal derivative of
temperature in the Neumann condition is approximated by the first-order
one-sided finite difference scheme, from which the wall temperature can be
computed. With the calculated wall temperature, the problem subjected to
Neumann condition is converted to a problem subjected to Dirichlet condition
where the explicit direct forcing method used in the work of Zhang & Zheng
(2007) for isothermal flows is applied to correct the predicted temperature
field to the corrected one. While many extra procedures and efforts are
required in their work, it is obvious that our proposed method is more efficient

and straightforward.



Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
134

5.1.3 Computational Sequence
The basic solution procedure of the proposed method can be outlined below:
1) Use the solution procedures described in Chapter 2 or 3 to compute the
velocity field
u
.
2) Solve Eq. (4.8) to get the predicted temperature
*
T
.
3) Use Eqs. (5.2)-(5.3) to calculate
()
*
,
i
T
t
n
∂
∂
X

),,1( Mi "
=
and then
substitute it into Eq. (5.4) to compute the boundary heat flux
()
,
i
QtΔ X
),,1( Mi "=
.
4) Calculate the heat source
),( tq
j
x
),,1( Nj "
=
using Eq. (5.5).
5) Correct the fluid temperature at Eulerian points using Eq. (4.7). Until now,
both the velocity field and temperature field have been updated to time
level
1+n
.
6) Repeat steps (1) to (5) until a desired solution is achieved.

5.2 Numerical Examples
The present boundary condition-implemented IBM, using velocity correction
and flux correction technique, will be validated in this section through its
application it to simulate both forced convection (forced convection over a
stationary isoflux circular cylinder) and natural convection (natural convection
in a horizontal concentric or eccentric cylindrical annulus between an inner

isoflux cylinder and an outer isothermal cylinder) problems. Before we solve

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
135

convection problems, we will use a model heat conduction problem to
investigate the spatial accuracy of the present solver.

5.2.1 Numerical analysis of spatial accuracy
The heat conduction problem in Section 4.3.1 is once again used as a model
example to investigate the spatial accuracy of proposed thermal IBM solver,
where the governing equation is described by Eq. (4.32) and the temperature
boundary conditions are specified as:
1
in
T
n
∂
=
∂
on the inner boundary and
(
)
22
1ln2
out
Txy=+ +
on the outer boundary. The problem is solved using
five different uniform meshes with mesh spacing of

111 1
,,,
16 32 64 128
h =
and
1
256
respectively. The spatial accuracy is measured by
1
L
norm of the
relative error, which is defined in the same way as in Griffith & Peskin (2005)
and Mori & Peskin (2008). Fig. 5.1 shows the relative
1
L
error of the
numerical solution with respect to the mesh spacing, which indicates a slope
of 2, implying the second order of spatial accuracy.

5.2.2 Forced convection over a stationary isoflux circular cylinder
Forced convective heat transfer from a stationary heated circular cylinder
which is immersed in a cold free stream and releases constant and uniform
heat flux is simulated for several low and moderate Reynolds numbers of

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
136

,10Re = ,20
40

and 100 and fixed Prandtl number of
7.0Pr
=
. The setups in
the present problem are exactly the same as those in subsection 4.2.1 except
the boundary condition is replaced by the specified uniform heat flux. Heat
transfer characteristics of isotherms, local Nusselt number distribution and
average Nusselt number on the cylinder surface are presented.

In the simulation, the temperature is normalized by

B
TT
T
QD
k
∞
−
′
=
(5.7)
where
∞
T
is the free stream temperature, and
B
Q
is the specified uniform
heat flux at the cylinder surface in its normal direction. The thermal condition
on the immersed boundary can then be expressed in the dimensionless form

1
T
n
′
∂
−=
∂
(5.8)

The local and average Nusselt numbers on the cylinder surface are defined as

()
()
()
()
c
hsD
Nu s
k
=
X
X
(5.9)
and

()
1
()
2
Nu Nu s ds

D
π
Γ
=
∫
X
(5.10)
where
(())
B
c
Q
hs
TT
∞
=
−
X
is the local convective coefficient. Their specific
dimensionless forms are

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
137


()
()
1
Nu s

Ts
=
′
X( )
X( )
(5.11)

()
2
0
11
2
Nu d
T
π
θ
πθ
=
′
∫
(5.12)
that is, local Nusselt number is exactly the reciprocal of local dimensionless
temperature. Therefore, for the problem of specified heat flux condition, the
surface temperature is one of the important variables in the calculation, since it
can reflect the heat transfer characteristics along the surface.

Fig. 5.2 shows isotherms in the vicinity of the cylinder for each case. As can
be observed from Fig. 5.2, the isotherms slightly cluster in the front surface of
the cylinder, indicating a larger temperature gradient, or a higher heat transfer
rate there than other regions. Furthermore, with an increase of Reynolds

number
Re
, the temperature around the cylinder surface is decreased. From
Eq. (5.12), we can say that the heat transfer is enhanced.

Table 5.1 lists a comparison of computed average Nusselt numbers for Re =
10, 20, 40 with reference data in the literature (Ahmad & Qureshi 1992;
Dhiman et al. 2006; Bharti et al. 2007). Fig. 5.3 draws the local Nusselt
number distribution on the cylinder surface along with the result of Bharti et al.
(2007) for Re = 10 and 20. All these results show a good agreement. Fig. 5.4
plots the time evolution of average Nusselt number for
100Re
=
, which,
once again, implies an obvious periodic variation of the flow field. As

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
138

expected, the average Nusselt number
Nu
on the cylinder surface
increases with Reynolds number
Re
.

5.2.3 Natural convection in a concentric horizontal cylindrical annulus
between an outer isothermal cylinder and an inner isoflux cylinder
The capability of present method is now tested by a natural convection

problem. Generally, natural convection is more complex than forced
convection since its velocity and temperature fields are strongly coupled. The
buoyancy force is the driving force for the flow, and the Boussinesq
approximation is often used. Here, natural convection in a horizontal
concentric cylindrical annulus is simulated. The schematic view for the
problem configuration is shown in Fig. 5.5, where the surface of the inner
cylinder with radius
i
R
is maintained at a uniform heat flux
B
Q
, and the outer
cylinder of radius
2
oi
RR=
is kept at a constant temperature
∞
T
. The flow
behavior of this problem is characterized by Prandtl number
P
r
, Rayleigh
number
23
/
pB
cgLQGk

Ra
k
ρβ
μ
= , where
oi
GR R
=
−
is the gap width of the
annulus. In this study, numerical investigations are carried out for three
Rayleigh numbers of
5700,1000 =Ra
and
4
105×
while Prandtl number is
kept at
Pr 0.7=
. The initial conditions are set as zero for
u
and
∞
T
for
T

in the whole computational domain. The gap width G is taken as the reference
length, and the temperature is normalized by Eq. (5.7). A uniform Eulerian


Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
139

mesh with resolution /64hG
=
and convergence criteria
16
110
nn+−
∞
−<×uu&& and
18
110
nn
TT
+
−
∞
−<×&& is used for all the studied
cases. Numerical results in terms of streamlines, isotherms and local
temperature distribution on the inner cylinder are displayed.

As expected, the flow and thermal fields are symmetric about the vertical
central line through the center of the annulus (Fig. 5.6). A pair of
crescent-shaped eddies are formed in the enclosure, one in each half. When
Rayleigh number is small (
3
10=Ra
), the heat flow in the enclosure is

conduction-dominated, and the isotherms appear as a series of concentric
circular-like shapes around the inner cylinder. When Rayleigh number is
increased (
4
5700, 5 10Ra =×
), buoyancy begins to play a more important role,
and the thermal boundary layer on the bottom surface of the inner cylinder
becomes thinner than that on its top surface. Meanwhile, strong convection
induces a plume on the upper part of the annulus. Also, it is seen that with an
increase of Rayleigh number, the plume becomes stronger and drives the flow
impinging on the top wall of the outer cylinder, leading to a thinner thermal
boundary layer and denser isotherm gradient around the surface of the inner
cylinder and top wall of the outer one. As a consequence, the heat transfer in
these regions is enhanced. These phenomena can be verified in Fig. 5.7, which
shows that maximum temperature on inner cylinder occurs at its uppermost
point and the minimum temperature appears at its lowermost point for all the

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
140

three cases. That is, at its lowermost point, the heat transfer rate is largest
while at its uppermost point, the heat transfer rate is smallest. Furthermore,
Fig. 5.7 reveals that the temperature at any location on the inner cylinder is
always higher for larger
Ra
as compared to smaller one, indicating that the
heat transfer rate increases with an increase of
Ra
.

The local temperature distributions on the inner cylinder surface are displayed
for
5700=Ra
,
4
105× in Fig. 5.8, while reference profiles in the literature
(Yoo 2003) are also included. Their comparison indicates that the results
obtained by present method agree well with the reference data.

5.2.4 Natural convection in an eccentric horizontal cylindrical annulus
between an outer isothermal cylinder and an inner isoflux cylinder
In this subsection, the proposed method is further tested by another natural
convection problem. The geometry of the problem under consideration (as
shown in Fig.5.9) is similar to the one investigated in Section 5.2.3, except
that the two infinite horizontal cylinders are eccentrically arranged in vertical
direction. The eccentricity of the inner cylinder is denoted by
e
, whose
positive value represents the upward direction. The fluid flow and heat transfer
of the problem are characterized by the Rayleigh number
Ra
, Prandtl
number
Pr
, radius ratio
/
oi
RR
and eccentricity

/eG
ε
=
, where
Ra
,
Pr

and
G
have the same definition as those in Section 5.2.3. In the present

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
141

study, numerical calculations are performed for four different Rayleigh
numbers of
3
10Ra =
,
4
10
,
5
10
,
6
10
with the other three parameters fixed

at
Pr 0.7=
,
/2.6
oi
RR=
and
0.625
ε
=
−
, corresponding to the numerical
study of Ho et al. (1989). Uniform Eulerian meshes with mesh size
/100hG= are used for all the considered simulations, while the convergence
criteria are set as
15
110
nn+−
∞
−<×uu&& and
18
110
nn
TT
+
−
∞
−<×&&.

Fig. 5.10 shows the streamlines and isotherms at various Rayleigh numbers. It

is observed that for this eccentric geometry, the annular gap at the top region
over the inner cylinder is enlarged, making the convective flow stronger there.
The qualitative features of isotherms illustrated above for the concentric
geometry appear to be even more pronounced. The local temperature profile
along the inner cylinder surface is depicted in Fig. 5.11 for different
Ra
. Also
included in the figure are the results of Ho et al. (1989) for the purpose of
comparison. As expected, the present results are in good agreement with those
of Ho et al. (1989). Finally, the average heat transfer rate over the inner
cylinder surface is examined, which is represented by means of average
Nusselt number defined as

()
B
in out
QG
Nu
kT T
=
−
(5.13)
The results obtained via Eq. (5.13) and corresponding data of Ho et al. (1989)
are displayed in Table 5.2. Obviously, the present results match fairly well

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
142

with reference data of Ho et al. (1989).

5.3 Conclusions
In this chapter, an efficient IBM is proposed for thermal flow problems with
heat flux boundary condition. The effect of thermal boundaries to the flow and
temperature fields is considered through velocity correction and heat flux
correction in the framework of immersed boundary method. In particular, a
heat source term, which is distributed from the offset heat flux at boundary via
Dirac delta function interpolation, is introduced into the energy equation.

The efficiency and capability of present method are validated by applying it to
simulate both forced convection and natural convection problems. Heat flow
characteristics in terms of local Nusselt number or temperature distribution are
presented. The good agreement between present results and available data in
the literature indicates that the present method provides a useful tool for
solving heat transfer problems with Neumann boundary conditions.

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
143



Table 5.1 Comparison of average Nusselt number
Nu
for
40,20,10Re=


References

Nu


Bharti et al. (2007) 2.0400 2.7788 3.7755
Ahmad et al. (1992) 2.0410 2.6620 3.4720
Dhiman et al. (2006) 2.1463 2.8630 3.7930
Present 2.0265 2.7413 3.7407


Table 5.2 Comparison of average Nusselt number
Nu

on the inner cylinder surface

References
Nu

3
10Ra =

4
10Ra =

5
10Ra =

6
10Ra =

Present

2.3418 3.2052 4.5309 6.8879
Ho et al. (1989) 2.2724 3.2071 4.4821 6.8942
40Re =
20Re
=
10Re
=

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
144





Fig. 5.1 the
1
L
-norm of relative error of the temperature versus the mesh
spacing for the model problem






slope 2.01
=


Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
145




















Fig. 5.2 Isotherms for flow over a heated stationary cylinder at different
Re

10Re
=
20Re

=
40Re
=
100Re
=

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
146

















Fig. 5.3 Comparison of local Nusselt number distribution on the cylinder
surface for
Re 10, 20
=

θ
θ

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
147





Fig. 5.4 Time evolution of average Nusselt number on cylinder surface for
100Re
=



Fig. 5.5 Schematic view of natural convection in a horizontal concentric
cylindrical annulus
∞
T
B
Q
G

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
148


Fig. 5.6 Streamlines (left) and isotherms (right) for different
Ra


1000
=
Ra
4
105×=Ra
5700
=
Ra

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
149


Fig. 5.7 Effect of Rayleigh number on local temperature distribution along the
inner cylinder surface




(a)
5700Ra
=






θ
θ

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
150


(b)
50000Ra
=

Fig. 5.8 Comparison of local temperature distribution on the inner cylinder
surface for
5700
=
Ra
and
4
105×




Fig. 5.9 Configuration of natural convection in an eccentric horizontal
cylindrical annulus

θ
e
i
R
o
R
o
T
B
Q
θ
r

Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux

Boundary Condition
151






Fig. 5.10 Streamlines (left) and isotherms (right) for different
Ra



Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux
Boundary Condition
152


Fig. 5.11 Comparison of temperature profile along the inner cylinder surface


θ