Development of immersed boundary methods for isothermal and thermal flows 7

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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
Incompressible Thermal Flows
215
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Chapter 7

Applications of Developed IBM Solvers to Simulate
Three Dimensional Incompressible Thermal Flows

In this chapter, the IBM solvers proposed in Chapters 2, 4 and 5 are applied to
numerically study three-dimensional incompressible thermal flows.
Specifically, the forced convective heat and mass transfer from stationary or
rotating spheres in uniform cross flow, and natural convective heat transfer
inside spherical annulus are studied.

The incompressible viscous thermal flow around spheres is a fundamental
fluid dynamic and heat transfer problem with widespread scientific and
engineering applications. Bioreactors, industrial fluidized beds, combustion
systems, and chemical processes, etc. are among the well-known examples. In
spite of its simple and axisymmetric geometry, the sphere always induces fully
three-dimensional flows which would admit complicated kinematics. The
forced convective heat transfer, as pointed out, is a passive scalar transport
governed by the flow field such that the thermal field should experience
corresponding variations. The natural convective heat transfer, on the other
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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216


hand, is induced by buoyancy due to the temperature gradients so that the fluid
and thermal fields interact intensively and are strongly coupled. In the present
study, the fluid behaviors as well as the heat transfer performances for both
forced and natural convective heat transfer are numerically investigated.
Specifically, forced convective heat and mass transfer around a single
stationary sphere, two tandem stationary spheres and a single streamwise
rotating sphere, as well as natural convective heat transfer inside concentric
and vertically eccentric spherical annuluses are simulated. The thermal
conditions on the sphere surface, in most cases, are set to be of Dirichlet type,
for the convenience of direct comparison and analysis. Heat flux condition is
only considered in a few particular cases. 

7.1 Forced convective heat and mass transfer around a
stationary isolated sphere
Knowledge concerning the flow around an isolated stationary sphere has been
well accumulated by experiments and numerical simulations during the past
decades. Just as its two-dimensional counterpart (flow around a circular
cylinder), the problem of flow around a stationary sphere has been considered
as an excellent case for validating new numerical methodologies in three
dimensions.

The flow field around a single sphere is recognized to enjoy rich transition
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217

modes, depending on the Reynolds number
Re

, which is frequently defined
as
Re
UD
ρ
μ
∞
=
based on the sphere diameter
D
and the uniform incoming
velocity
U
∞
. At low Reynolds numbers, an axisymmetric vortex ring is
formed behind the sphere and attached to its surface. The flow is steady and
topologically similar at various
Re
. With an increase in Reynolds number,
the vortex ring downstream of the sphere shifts off-axis and the flow no longer
exhibits axial symmetry. Although non-axisymmetric, the flow does, however,
contain a plane of symmetry and remains steady. When the Reynolds number
increases further, a third transition mode occurs, at which stability of vortex
ring is lost. The flow is now unsteady but periodic. The vortex is shed from
the sphere surface with a single dominating frequency. A continued increase in
the Reynolds number would lead to very complex flow behaviors and is
beyond our investigation. These abundant phenomena and the critical
Reynolds numbers at which the transition modes occur have been explored by
many researchers. Taneda (1956), using the flow visualization method,
identified that the generation of an axisymmetric vortex ring occurred

at
Re 24≈
. Magarvey & Bishop (1961), through the dye visualization, found
that the wakes behind the liquid spheres exhibited the same vortex structure as
that observed by Taneda (1956). Besides, he noticed that the stable and
axisymmetric rings persisted up to
Re 210
=
but developed into a
non-axisymmetric pattern characterized by two parallel threads in the range
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218

of
210 Re 270<<
. This double-thread, beyond
Re 270
=
, lost its stability
and shed from the sphere. Numerically, by employing a spectral element
method, Tomboulides (1993) predicted the initial separation at
Re 20=
and a
transition from axisymmetric to double-thread wakes at
Re 212=
. The
stability of the axisymmetric flow was also examined by Natarajan & Acrivos
(1993) using a finite-element method, who suggested a regular bifurcation

at
Re 210=
. Johnson & Patel (1999) investigated the flow regimes at
Reynolds numbers up to 300 both numerically and experimentally. The lower
and upper Reynolds number limits for steady axisymmetric regime was
reported to be 20 and 210 while the onset of a periodic vortex shedding flow
was recorded around
Re 270=
. There are still many other excellent efforts on
the flow characteristics investigations, including the shedding frequency
measurement (Achenbach 1974; Kim & Durbin 1988; Sakamoto & Haniu
1990), vortex structure visualization (Shirayama 1992), etc.

The convective heat transfer from an isolated sphere has been the subject of
extensive investigations, as summarized by Clift et al. (1978) and Polyanin et
al. (2002). Ahmed & Yovanovich (1994) proposed an approximate analytical
solution in the range of Reynolds number
4
0Re210≤≤×
based on
linearization of the energy equation. Whitaker (1972), by examining his
experimental data, provided a correlation for the average Nusselt number in a
wide range of Reynolds numbers
5
1Re10≤≤
. Numerically, Dennis et al.
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219


(1973) calculated the heat transfer from an isothermal sphere at low values of
Reynolds numbers up to 20. Dhole et al. (2006) investigated the heat transfer
characteristics in the steady symmetric flow regime for both the constant
temperature and constant heat flux boundary conditions on the solid sphere
surface.

The convective heat and mass transfer from an isolated stationary sphere is
governed by two characteristic parameters: the Reynolds number
Re
and the
Prandtl number
Pr
. In the current simulation,
Pr
is fixed at 0.71 and
Re
is
taken as 100, 200, 250 and 300, which covers all the three flow regimes
identified in the literature. For the steady axisymmetric case (
Re 100=
and
200), both kinds of thermal boundary conditions have been considered on the
sphere surface, due to the availability of published results for convenient
comparison. They are set as: the isothermal condition
1
B
T
=
and the isoflux

condition
1
T
n
∂
−=
∂
. As for cases in the other two flow regimes (
Re 250=

and 300), calculations are carried out for the isothermal condition (
1
B
T =
)
only. A computational domain of size 25 20 20DDD
×
× is used, with the
sphere located at ( 15D , 10D , 10D ). The domain is discretized by a
non-uniform mesh with a fine resolution of /40hxyzD
=
Δ=Δ=Δ= around
the sphere. For the unsteady flows, a time step size of
0.001t
Δ
= is selected.

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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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220

7.1.1 Steady axisymmetric flow regime
In a certain low Reynolds number range, a steady axisymmetric vortex ring is
reported to form behind the sphere (Taneda 1956; Tomboulides 1993;
Magnaudet et al. 1995), which is often referred to as the “steady axisymmetric
flow regime”. Two Reynolds numbers of
Re 100
=
and 200 are selected for
the simulation and description of flow behaviors. The three-dimensional
vortex structures identified using the
2
λ
-definition proposed by Jeong &
Hussain (1995) are plotted in Fig. 7.1(a)-(b), where the axisymmetry is clearly
visualized, being toroidal and concave toward the sphere tail (without special
illustration, the three-dimensional vortex structures in the following are all
identified using the method of Jeong & Hussain (1995)). Streamlines in the
(, )
x
y
-plane at both Reynolds numbers are presented in Fig. 7.2. As can be
observed, the flow separates from the surface of the sphere and then rejoins,
forming a closed separation bubble which attaches to the sphere surface. It is
also noted that the flows in both planes are symmetric about the centerline,
and their topologies keep identical. Variations exist only in the separation
location, position of the vortex center and length of the recirculation region.
With Reynolds number increasing from
Re 100

=
to 200, the separation point
on the sphere surface moves towards the front stagnation point while the
vortex center extends downstream and the recirculation region becomes
stretched.

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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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221

The steady-state thermal fields are plotted in terms of temperature contours
(isotherms) in Figs. 7.3 and 7.4, for isothermal condition and isoflux condition,
respectively. As expected, the isotherms are symmetric about the centerline for
both types of thermal condition. They cluster heavily around the front surface
of the sphere, indicating a large heat transfer rate there. As the Reynolds
number increases from 100 to 200, the clustering of isotherms on the front
surface shows some enhancement. Moreover, the isotherms around the back
stagnation point which is thinly distributed for
Re 100
=
, are more densely
spaced at
Re 200
=
. These observations can be further verified in Fig. 7.5,
where the local Nusselt number distribution along the sphere surface is
presented (the angle
θ
is measured in the clockwise direction from the front

stagnation point (
0
θ
°
=
) to the rear one (
180
θ
°
=
)). Each curve manifests two
peaks located around the front and rear stagnation points, with the former
much higher than the latter. The Nusselt number throughout the sphere surface
is enhanced as
Re
increases from 100 to 200. The profiles reported by Dhole
et al. (2006) for both boundary conditions are included in Fig. 7.5 as well,
which show a good agreement with ours.

The present results are also quantitatively validated in Table 7.1 by making a
comparison of our calculated drag coefficient with the numerical
measurements of Johnson & Patel (1999), Gilmanov et al. (2003) and White
(1974). Note that for all the problems in this chapter, the drag coefficient is
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
Incompressible Thermal Flows
222

defined as

2
2
1
24
D
D
F
C
D
U
π
ρ
∞
=
, (7.1)
where

D
F
is the drag force acting on the sphere. The comparison shows that
our results in the steady axisymmetric flow regime match well with those in
the literature.

A further validation is implemented by examining the heat transfer rate on the
sphere surface. For the isothermal case, Table 7.2 lists the surface-averaged
Nusselt number
Nu
obtained from the present results and those calculated
from the published correlations in the literature. Our numerical value is
calculated by taking an area-integral of the local Nusselt number over the

sphere surface and then making an average in the way

2
1
4
Nu Nu ds
D
Γ
π
=
∫
i

(7.2)
where
Γ
denotes the sphere surface. We can see that our results agree
reasonably well with the correlations reported by Ljachowski (1940),
Froszling (1938), Whitaker (1972) and Feng & Michaelides (2000)

7.1.2 Steady planar-symmetric flow regime
The flow at
Re 250=
is taken as a representative to illustrate the flow
features in the steady planar-symmetric flow regime. Streamlines in both
(, )
x
y
-plane and
(,)

x
z
-plane are presented in Fig. 7.6, which clearly reveals
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223

the loss of the axial-symmetry in the current flow regime. Although
non-axisymmetric, the flow remains steady and exhibits symmetric in the
(,)
x
z
-plane. It is also noted that the rear stagnation point in the
(, )
x
y
-plane
moves forward along the sphere surface and stays away from the centerline.
The three-dimensional vortex structure is plotted in Fig. 7.1(c) where the
vortex behind the sphere is different from the toroidal structure of
axisymmetric case, and is developed into a double-threaded structure in the
planar-symmetric flow regime.

While the isotherms in the
(,)
x
z
-plane still keep symmetric about the
centerline, the onset of non-axial symmetry, as expected, results in an

asymmetric behavior of thermal field in the
(, )
x
y
-plane (Fig. 7.7). The local
Nusselt number along the sphere surface in the
(,)
x
z
-plane is depicted in Fig.
7.8. As compared to the axisymmetric cases (
Re 100
=
and 200), the heat
transfer is enhanced on the surface around the front stagnation point. In the
(, )
x
y
-plane (Fig. 7.8(b)), the heat transfer rate on the front hemisphere
(
0100
θ
≤≤

and
260 360
θ
≤≤

) is coincident with that in the

(,)
x
z
-plane, showing a symmetric behavior. On the rear hemisphere,
anti-symmetry happens and the local peak no longer appears at
180
θ
=

but
moves forward to
170
θ
=

. Additionally, the peak value increases as
compared to that in the
(,)
x
z
-plane.

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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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224

The fluid and thermal behaviors at
Re 250
=

, in forms of average drag
coefficient and Nusselt number, are compared with those established ones in
Tables 7.3 and 7.2 respectively, from which good agreements are observed.

7.1.3 Unsteady periodic flow regime
As
Re
is increased above approximately 270, instability becomes so
pronounced that flow unsteadiness is triggered. In this periodic unsteady flow
regime, flow at a Reynolds number of 300 was chosen as the case of interest.
The time evolution of drag coefficient and surface-averaged Nusselt number
on the sphere is well traced and plotted in Fig. 7.9, from which a perfect
periodic characteristic is observed, showing that the vortex is shedding
periodically from the sphere. The vortex shedding frequency is frequently
described by the dimensionless Strouhal number, which, based on our
calculation, is 0.133 and basically agrees well with the published value of
0.136 reported by Tomboulides (1993), and 0.137 provided by Johnson &
Patel (1999).

The instantaneous in-plane streamlines in one vortex shedding cycle are
plotted in Figs. 7.10 and 7.11, corresponding to four equal-interval phases. It
is clear from the pictures that the streamlines in the
(,)
x
z
-plane (Fig. 7.10)
remain symmetric throughout the cycle, indicating that the observed
plane-symmetry for the steady flows is still present in the unsteady ones,
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D

Incompressible Thermal Flows
225

which agrees with the findings of Johnson & Patel (1999) and Kim & Choi
(2002). The movement of the recirculation region as revealed in Johnson &
Patel (1999), is also obviously observed in this work. Streamlines in the
symmetry plane, i.e.
(, )
x
y
-plane, as observed in Fig. 7.11, show a sequence
of patterns similar to that of the steady case (
Re 250
=
).

The three-dimensional wake structures in this flow regime become more
complex. As shown in Fig. 7.1(d), the vortices are no longer attached to the
sphere but are seen to shed regularly from the sphere surface and develop into
a pair of legs behind it. The hairpin vortex structure, which has been reported
by Kim & Choi (2002) and Giacobello et al. (2009), is nicely captured in the
present work.

Consistent with the streamlines in each phase, the isotherms in the
(,)
x
z
-plane (Fig. 7.12) are symmetric about the centerline throughout the
vortex shedding cycle and evolve following the motion of fluid flow. Their
distributions in the vicinity of the sphere surface, however, are almost

independent to time. Isotherms near the sphere surface in the
(, )
x
y
-plane
(Fig. 7.13) do not change with time either. They always concentrate around the
stagnation points, implying that the stagnation points do not change with time.
The above visualizations can be further clarified by the Nusselt number
profiles plotted in Fig. 7.14, from which it is observed that the Nusselt number
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226

in the
(,)
x
z
-plane (Fig. 7.14(a)) are symmetric about
180
θ
°
=
for all the
four phases. They peak at the front and rear stagnation points of
0
θ
°
=

and
180
θ
°
=
, with the former much larger than the latter. Except for the
symmetric characteristic, the Nusselt number profiles in the
(, )
x
y
-plane (Fig.
7.14(b)) show similar behaviors. However, the second peak on the sphere
surface no longer exhibits itself at
180
θ
°
=
but moves forward to
160
θ
°
=

approximately. Now we can see that while the front stagnation point always
resides at
0
θ
°
=
, the rear stagnation point in the

(, )
x
y
-plane keeps moving
forward from
180
θ
°
=
at
Re 100
=
and 200, to
170
θ
°
≈
at
Re 250=

and further to
160
θ
°
≈
at
Re 300
=
.

The time-mean drag coefficient
D
C
and Nusselt number
Nu
, which are
defined as

1
shed
DD
shed
T
CCdt
T
=
∫
(7.3)

1
shed
shed
T
Nu Nu dt
T
=
∫
(7.4)
where
s

hed
T
is the vortex shedding period, are presented in Table 7.4 and 7.2
respectively, and compared with the published results. Once again, good
agreement is achieved.

From the above simulation results, we can see that as the Reynolds number
increases, the flow instability increases correspondingly and the flow develops
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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227

from steady axisymmetric to unsteady plane symmetric. While the drag force
experienced by the sphere successively decreases, the overall heat transfer
from the sphere surface is monotonically enhanced.

7.2 Forced convective heat and mass transfer around a pair of
tandem spheres
As demonstrated in the above subsection, complex flow behaviors can be
produced from one isolated sphere in a free stream. It is believed that flow
phenomena would become more abundant and interesting when two identical
spheres are located in tandem, due to the dynamic interaction between the
wakes behind each sphere. Consequently, flow past a pair of tandem spheres
attracts quite a number of researchers.

Pioneering studies, such as the ones conducted by Rowe & Henwood (1961),
Lee (1979), Tsuji et al. (1982), were mainly dedicated to the direct
measurement of drag forces on interacting spheres. Later, Tal et al. (1984)
investigated both the hydrodynamic and thermodynamic interaction of two

tandem spheres in a steady uniform flow at
Re 40
=
. Zhu et al. (1994)
examined the effect of separation distance and
Re
on the drag forces for
Re

ranging from 20 to 130. Prahl et al. (2007) reported the variation of the drag
and lift forces for Reynolds number of 50, 100 and 200. Yoon & Yang (2007),
by performing a parametric study, estimated flow-induced forces on two
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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228

arbitrarily positioned spheres at
Re 300
=
. All these studies found that the
drag forces on both spheres were always less than that of an isolated one and
that the reduction was much more significant for the downstream sphere.
Other than drag forces, more recent works have also concentrated on
examining the wake structure and its effect on dynamic forces. Zou et al.
(2005) paid their attentions to the effect of separation distance on the flow
patterns at
Re 250
=
and predicted three different flow regimes. Following

Zou’s work, Prahl et al. (2009) investigated the force characteristics and
shedding patterns under the influence of separation spacing at
Re 300=
. The
aforementioned studies clearly indicate that for two spheres in tandem
arrangement, the Reynolds number
Re
UD
ν
∞
=
(
U
∞
is velocity of the
uniform incoming flow) and their separation distance
G
(
G
is the
center-to-center distance between the spheres) are the two most important
parameters. The drag, lift forces and wake patterns strongly depend on them.

In this subsection, forced convective heat and momentum transfer around a
pair of tandem spheres are numerically studied at two Reynolds numbers: a
low one of 40 and a moderate one of 300. Two separation distances of
/1.2GD=
and 2.5 are simulated for
Re 40
=

, and three separation distances
of
/1.5GD=
, 2 and 3 are considered for
Re 300
=
. Their flow patterns,
drag and lift coefficients and heat transfer performances are well analyzed. A
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
Incompressible Thermal Flows
229

computational domain of size (25 ) 20 20DG D D
+
×× is chosen such that the
inlet is located
10D ahead of the upstream sphere and the outlet 15D
behind the downstream sphere, with the spheres located at (
10D ,10D ,10D)
and (
10DG+ ,10D ,10D ). A non-uniform mesh is used for the domain
discretization, with a locally fine resolution of / 40
hxyzD
=
Δ=Δ=Δ=
covering the two spheres. For the unsteady cases, a step size of
0.001tΔ= is
selected for the time integration.

7.2.1 The case at
Re 40=

The flow field at this low Reynolds number is always axisymmetric regardless
of the separation distance, as seen from the three-dimensional vortex
structures in Fig. 7.15. The steady state streamlines plus isotherms in the
(,)
x
z
-plane are depicted in Figs. 7.16 and 7.17 for
/1.2GD
=
and 2.5,
respectively. It is observed that the recirculation region behind the upstream
sphere for
/1.2GD=
(Fig. 7.15) contacts the downstream sphere while it
does not happen in the case of
/2.5GD
=
, which, as a result, are frequently
referred as the contacting case and non-contacting case, respectively. In the
contacting case, the downstream sphere is located so close to the upstream one
that the stagnation-type flow in front of it is replaced with a recirculation zone,
and the dense concentration of isotherms, which is supposed to be observed on
the front surface of the sphere, does not show up either. The existence of the
recirculation zone corresponds to a low pressure region and results in a
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
Incompressible Thermal Flows

230

considerable reduction of drag. In the non-contacting case, the recirculation
zone behind the upstream sphere is being stretched as compared to that of a
single sphere. Although the stagnation point and isotherms gathering reappear
at the front surface of the downstream sphere, its concentration, i.e.,
temperature gradient, is clearly weaker. For both cases, flow separation occurs
earlier on the upstream sphere and much later on the downstream one (for easy
comparison, the streamlines for an isolated sphere is plotted in Fig. 7.18). The
recirculation region behind the downstream sphere, at the same time,
significantly contracts itself. These observations are in well agreement with
those of Tal et al. (1984) and Zhu et al. (1994).

The local Nusselt number distribution along the sphere surface in its
circumferential direction is given in Fig. 7.19 for the contacting case
of
/1.2GD=
. The result for the case of isolated sphere is also provided for
convenience.
θ
is measured in the same way as previously. It is noted that
the distribution of local Nusselt number
Nu
on the upstream sphere has not
been changed as compared to the case of isolated sphere up to the location
around
80
θ
°
=

, after which, large discrepancy is captured and
Nu
on the
upstream sphere drops drastically. This is due to the close contact of the
downstream sphere such that there is no enough room for the entrainment of
the cold free-stream fluid. The local Nusselt number on the downstream
sphere is significantly affected by the presence of upstream one. A
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231

considerable reduction is produced on its front surface and a maximum occurs
at around
100
θ
°
=
where the isotherms are concentrated, which should
approximately correspond to the separation location.

The local Nusselt number distributions for the non-contacting case of
/2.5GD=
are plotted in Fig. 7.20. On the upstream sphere, the large
reduction in Nusselt number on its leeward side does not occur, as compared
to the contacting case. It seems that the local Nusselt number almost recovers
to that for the isolated case. On the downstream sphere, the local heat transfer
rate is enhanced throughout the entire surface as compared to the contacting
case, especially in the leeward side (approximately from
110

θ
°
=

to
180
θ
°
=
).

A quantitative comparison of the mean drag coefficient and Nusselt number is
made between our results and the experimental measurements of Rowe et al.
(1961) and numerical simulations of Tal et al. (1984) and listed in Table 7.5.
While the present drag coefficients are a little larger than the reference values,
the Nusselt numbers agree quite well with those from Tal et al. (1984). The
results show that, the drag forces on both spheres are increased as the
separation distance
/GD
increases from
1.2
to
2.5
and the heat transfer
rates are enhanced for both spheres.

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232


7.2.2 The case at
Re 300=

When the Reynolds number is increased to
Re 300
=
, the flow patterns will
transit from one mode to another depending on the separation distance. The
present study considers three separation distances of
/1.5GD
=
, 2.0 and 3.0,
corresponding to three different flow regimes: steady axisymmetric, steady
plane symmetric and unsteady plane symmetric.

Our calculations show that the flow at small separation distances of
/1.5GD=
and 2.0 would eventually reach a steady state. While a
toroidal-structured vortex (Fig. 7.21(a)) is enveloping the spheres
at
/1.5GD=
, the vortex at
/2.0GD
=
loses its axisymmetry and is
characterized by two parallel threads behind the downstream sphere
(Fig.7.21(b)). Recalling the flow pattern of an isolated sphere at the same
Reynolds number, it seems that the placement of a second sphere in the wake
can suppress the three-dimensional instabilities to a certain extent. Fig. 7.22

plots the steady-state streamlines and isotherms in the
(, )
x
y
-plane
for
/1.5GD=
, while the corresponding pictures for
/2.0GD
=
are shown
in Fig. 7.23 in both the
(, )
x
y
- and
(,)
x
z
-plane. Consistent with the
three-dimensional wake structures in Fig. 7.21(b), a breakdown of the flow
symmetry in the
(, )
x
y
-plane has been captured as the separation distance
increases from
/1.5GD=
to 2.0. In both cases, it is observed that the
recirculation region behind the upstream sphere is confined in the gap region

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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
Incompressible Thermal Flows
233

and closely contacts with the downstream sphere, and the stagnation-type flow
in front of the downstream sphere is therefore replaced with this recirculation
zone.

The isotherms in Fig. 7.22 for
/1.5GD
=
show weaker gathering around
the rear stagnation point of
180
θ
°
=
on the upstream sphere and disappearing
of dense concentration around
0
θ
°
=
on the downstream sphere, as compared
to the single isolated case (Fig. 7.3). This is due to the close contact of the
downstream sphere so that no room is available to allow the entrainment of
cold free-stream fluid. Nevertheless, the isotherms do exhibit relatively heavy
cluster around
70

θ
°
=
and
290
θ
°
=
on the downstream sphere, resulting in
peaks of local Nusselt number in Fig. 7.24. Recalling the streamlines in Fig.
7.22, these peaks approximately correspond to the reattachment points. The
thermal field for
/2.0GD=
is not axisymmetric, as seen from Fig. 7.23
where the temperature contours are symmetric in the
(,)
x
z
-plane and
asymmetric in the
(, )
x
y
-plane. In the
(,)
x
z
-plane, the isotherms around the
base (
180

θ
°
=
) of the upstream sphere and the front (
0
θ
°
=
) of the
downstream sphere are more concentrated than the former case of
/1.5GD=
,
as clearly revealed from the Nusselt number distribution in Fig. 7.24.
Meanwhile, the heat transfer rate around the two reattachment points on the
downstream sphere is enhanced as well. Regarding the isotherms in the
(, )
x
y
-plane, their concentration around the stagnation points is nicely
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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234
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captured. Although difference in the Nusselt number distribution does exist
between the two enumerated planes, the locations and values of the maximum
are coincident (Fig. 7.25) with each other.

At a separation distance of
/3.0GD

=
, the flow becomes unsteady but the
vortices are shed off regularly from the spheres. Zou et al. (2005) detected an
unsteady plane symmetric flow in their simulation, while Prahl et al. (2009)
reported that their flow at this separation distance presented a statically
axisymmetric behavior. To make a clear visualization of flow development at
this unsteady case, the perspective and side view of the wake structure are
presented in one cycle in Fig. 7.26. A careful observation of these pictures
shows that our results more closely resemble the findings of Zou et al. (2005),
presenting an unsteady plane-symmetric behavior. The
(, )
x
y
-plane continues
to present as a symmetry plane during the flow evolution process.

The local Nusselt numbers along the sphere surfaces in the
(,)
x
z
-plane and
(, )
x
y
-plane corresponding to Fig. 7.26 are shown in Fig. 7.27 (the angle
θ

is defined in the same way as before). The Nusselt number profiles on both
spheres in the
(,)

x
z
-plane are symmetric about
180
θ
°
=
, revealing the
in-plane symmetry of the isotherms. This observation matches well with the
temperature visualization in Fig. 7.28 (only the first-quarter phase is presented
as a representative). Meanwhile, all of the profiles in Fig. 7.27 maximize
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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235

around the front stagnation point of
0
θ
°
=
and present a local peak around
the rear stagnation point of
180
θ
°
=
. On the downstream sphere, the Nusselt
number exhibits a double-hump profile and maximizes at the two humps
corresponding to

65 ~ 70
θ
°°
=
and
290 ~ 295
θ
°°
=
respectively. This pair
of humps changes their locations and values slightly during one cycle.
Additionally, at
0
θ
°
=
and
180
θ
°
=
, two local peaks which are about
one-third as large as the maximum value are also presented.

The Nusselt number profiles in the
(, )
x
y
-plane share many of thermal
behaviors as described above except for the symmetry. On the upstream sphere,

the maximum heat transfer rate exhibits at the front stagnation point and keeps
the same throughout the cycle. Meanwhile, a local peak is always captured
around the rear stagnation point. On the downstream sphere, the Nusselt
number profiles show a double-hump pattern. The two humps are not of equal
height, with one higher than another. The locations of the two humps are
shifted periodically with time, due to the periodic formation and shedding of
the vortices between the two spheres.

The reliability of the present results is further confirmed in Table 7.6 by
making a quantitative comparison of the obtained force coefficients with those
reported by Yoon et al. (2007) and Prahl et al. (2009). It is observed that the
drag coefficients on the upstream sphere in the present study are somewhat
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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236

larger than the reference ones, but basically our results agree well with
reference data. Different from the low Reynolds number case at
Re 40=
,
with an increase in separation distance, the drag force at
Re 300
=
decreases
on the upstream sphere but increases on the downstream sphere. Nevertheless,
the drag forces on both spheres are reduced as compared to the isolated case.
The heat transfer rate, as listed in Table 7.7, however, increases monotonically
on both spheres as the separation distance increases from 1.5 to 3.0.

7.3 Laminar flow past a streamwise rotating isothermal sphere
Flow over a rotating sphere is of fundamental importance in numerous
engineering applications involving particle-particle or particle-wall collision.
Its flow characteristics have been recognized to rely significantly on the
direction of rotation. In the case of streamwise rotation where the sphere
rotates in the same direction as the incoming flow, Schlichting (1979), based
on the work of Luthander & Rydberg (1935) and Hoskin (1955), summarized
two specific features related to the variation of drag and separation line due to
the rotation. By means of numerical simulation, Kim & Choi (2002) shed light
on the modifications made by the streamwise rotation on the vortex structures
behind the sphere. Reynolds numbers of 100, 250 and 300, which cover the
axisymmetric steady, non-axisymmetric steady and vortex shedding regimes
for a stationary sphere were considered. Their simulation results exhibited
strong dependence of flow features on both the Reynolds number
Re
and the
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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237

rotational speed

Ω
. In this section, the wake transitions identified in Kim &
Choi (2002) are revisited in a forced convective laminar flow, in which a
streamwise rotating sphere with uniform high temperature is placed in a cold
incoming flow. Results for some representative Reynolds numbers and
rotational speeds, which are defined in the same way as in Kim & Choi (2002),
are presented and compared with Kim & Choi (2002). Moreover, the effect of

streamwise rotation on the heat transfer rate is examined. A finite domain of
size
25 20 20DDD×× is used as the computational domain, with the rotating
sphere located at (
10D ,10D ,10D). A non-uniform mesh is employed for its
discretization, with a locally fine resolution of / 40
hxyzD
=
Δ=Δ=Δ=
around the sphere. Meanwhile, a step size of
0.002t
Δ
= is used for the time
integration.

Fig. 7.29 shows the three-dimensional vortex structures for
0.3
Ω
=

and
1.0
Ω
=
at
Re 100=
. It is observed that they are both axisymmetric,
which is consistent with the observations of Kim & Choi (2002) that the wake
structures for all the rotational speeds
01

Ω
≤
≤
at this Reynolds number
remain steady and axisymmetric. However, as compared to the toroidal vortex
structure for the stationary case (
0
Ω
=
) in Fig. 7.1(a), the vortex for the
rotating cases comprises of a shroud over the sphere and a threaded structure
in the near wake. The streamlines in Figs. 7.30 and 7.31 demonstrate that
with

Ω
increasing from 0.3 to 1.0, the vortex becomes elongated and
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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238

stronger. On the other hand, the contour lines of temperature, as the rotational
speed is increased from 0.3 to 1.0, show an obvious tendency to gather
towards the base region of the sphere. This behavior is clearly reflected in Fig.
7.32. Observing the Nusselt number on the sphere surface at different
rotational speeds, it is found that while there is almost no difference between
the stationary case and the low-speed rotating case of
0.3
Ω
=

, a tangible
increase in
Nu
at the base region from
135
θ
°
=
to
180
°
is noted in the high
rotating speed case of
1.0
Ω
=
.

The instantaneous vortex structures for
Re 250
=
at rotational speeds
of
0.1
Ω
=
, 0.3 and 1.0 are presented in Fig. 7.33. It has already been known
that for a stationary sphere (
0
Ω

=
), the flow is steady and a pair of vortices
with the same vorticity strength but opposite direction appears in the wake
(Fig. 7.1(c)). When the sphere is under streamwise rotation, the flow becomes
unsteady. At
0.1
Ω
=
(Fig. 7.33(a)), because of the introduction of positive
streamwise vorticity from the sphere rotation, the aforementioned balance and
symmetry have been broken and one vortex is strengthened while the other is
weakened. This effect is enhanced at
0.3
Ω
=
(Fig. 7.33(b)), where the
weakened vortex disappears and the structure gives way to “single-threaded”.
As the rotational speed increases to
0.5
Ω
≥
(Fig. 7.33(c)), the
“double-threaded” structure is induced once again and the two vortices are
twisted together forming a complex pattern.
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Chapter 7 Applications of Developed IBM Solvers to Simulate 3D
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239

The time evolutions of the drag and lift coefficients are plotted in Fig. 7.34.

Apparently, the flows are unsteady at all the three rotational speeds. It is seen
that at small rotational speed of
0.1
Ω
=
(Fig. 7.34(a)) and 0.3 (Fig. 7.34(b)),
the two lift components

Ly
C
and
L
z
C
exhibit sinusoidal variations. The
magnitudes of the drag and lift, however, are constants and independent of
time, indicating that the vortex structures created by the rotating sphere are in
a frozen state. At the larger rotational speed of
1.0
Ω
=
(Fig. 7.34(c)), the
magnitudes of the drag and lift vary regularly with time, illustrating that the
flow is unsteady asymmetric and the strength of the vortices in the wake keeps
varying in time.

Due to the complexity of the flow structure as well as its evolution process, it
is difficult to provide a clear and full visualization of the thermal field.
However, the thermal field evolution, undoubtedly, strictly follows the
development of the flow field. In Fig. 7.35, a preliminary view regarding the

effect of sphere rotating on the heat transfer rate is provided by tracing the
surface-averaged Nusselt number
Nu
. Consistent with the drag coefficient,
the surface-averaged Nusselt number remains constant for
0.1
Ω
=
and 0.3
throughout the rotating process while it varies regularly at
1.0
Ω
=
.
Furthermore,
Nu
increases with an increase in
Ω
, showing that the
sphere rotation does produce some enhancement on the heat transfer rate.