Development of immersed boundary methods for isothermal and thermal flows 6

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


Chapter 6

Applications of Developed IBM Solvers to Simulate
Two-Dimensional Fluid and Thermal Flows

In this chapter, the fluid IBM solver proposed in Chapter 3 and the two
thermal IBM solvers proposed in Chapters 4 and 5 are numerically examined
by studying a collection of two-dimensional fluid and thermal flow problems.
The unsteady insect hovering flight which undertakes a harmonic translational
and rotational motion is simulated in Section 6.1. Section 6.2 discusses
particle sedimentations through vertical channels. In Section 6.3, the forced
convective heat transfer of a transversely oscillating cylinder in the wake of an
upstream cylinder is investigated.

6.1 Unsteady insect hovering flight at low Reynolds numbers
Flying insects in nature share plenty of fantastic aerodynamic performances
and maneuverabilities like taking off backward, flying sideways, and landing
upside down (Nachtigall 1974; Collett & Land 1975a, 1975b; Dalton & Kings
1975), which even our state-of-the-art vehicles are not capable of achieving.
Enchanted by the brilliant flight behaviors, researchers have endeavored to
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
explore the mechanisms behind it. One major effort is on how the flapping
wings generate forces (e.g. to support insects hovering in the air). Early
quasi-steady analysis, pioneered by Weis-Fogh & Jensen (1956) and
Weis-Fogh (1973), assumed that the force generation relied solely on the
instantaneous velocity and angle of attack while totally ignoring the past
history influence of wing motion, thus failing to provide sufficient force
required for hovering (Ellington, 1984). Recent studies recognized that three
distinct but interactive unsteady mechanisms: dynamic stall, rotational
circulation and wake capture, were responsible for the enhanced aerodynamic
performance of insects (Ellington et al. 1996; Dickinson & Gätz 1993;
Dickinson 1994; Dickinson et al. 1999; Sane & Dickinson 2001, 2002; Birch
& Dickinson 2001).

By measuring the time-dependent aerodynamic forces on an impulsively
started aerofoil within the Reynolds number range for flies of the genus
Drosophila and other small insects, Dickinson & Gätz (1993) indicated for the
first time that, the lift was largely enhanced during the translational motion of
the wing due to the presence of a leading edge vortex (or dynamic stall vortex).
Ellington et al. (1996) conducted smoke-visualizations on wings of a tethered
hawkmoth Manduca sexta and discovered that this leading edge vortex (LEV)
remained attached to the wing surface during translational motions in both the
up- and down-stroke and the high lift benefited from it maintains for the entire
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
stroke. By analyzing the momentum imparted to the fluid by the vortex, the
researchers believed that this must be a new high lift mechanism and named it

as delayed stall or dynamic stall mechanism (Ellington et al. 1996; Dickinson
et al. 1999). The LEV attachment, together with the large lift preservation, was
also observed by Birch and Dickinson (2001) for a fruit fly model Drosophila
melanogaster whose Reynolds number is high. Dickinson et al. (1999)
performed force measurements on wings of a dynamically scaled model of the
fruit fly Drosophila melanogaster and noticed the existence of transient peaks
in lift at each half-stroke reversal. Dickinson et al. (1999) suggested that two
mechanisms were responsible for the force peaks: wake capture mechanism at
the beginning of the stroke due to large effective fluid velocity and rotational
circulation mechanism at the end of the stroke due to rapid pitching-up
rotation.

Apart from valuable studies using aerodynamic models, computational fluid
dynamic analysis based on fluid-structure interaction has been developed as a
complementary method for the unsteady aerodynamics analysis. By using the
pseudo-compressibility method (Liu et al. 1995), Liu et al. (1998) confirmed
the delayed stall mechanism during the hovering flight. Sun and Tang (2002a),
by employing the artificial compressibility algorithm (Rogers and Kwak 1990),
discovered the close association between the force peaks and the translational
acceleration. After realizing that there is no evident spanwise flow at
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
low
2
Re 10∼
(Birch & Dickinson 2001), Wang et al. (2004) compared the
forces between two-dimensional computations and three-dimensional

experiments for several qualitatively different kinematic patterns (at low
Re

ranging from 75 to 115). They found that the results could achieve a
satisfactory match especially in advanced rotation and symmetrical cases. This
indicates that at low
Re
,
two-dimensional calculations could give reasonable
results. In the literature, a number of works have employed the
two-dimensional approach in the insect flight study (Gustafson & Leben 1991;
Wang 2000a; Wang 2004).

In this section, the two-dimensional unsteady flow around an elliptical
flapping wing which undergoes a prescribed harmonic translational and
rotational motion is simulated, to mimick insects’ hovering flight. Two most
common hovering modes, normal hovering (Fig. 6.1) and dragonfly hovering
(Fig. 6.2), are studied. Their governing equations can be written in a unified
form as

cos(2 / )cos
flap
XA tT
π
ψ
=
(6.1)

cos(2 / )sin
flap

YA tT
π
ψ
=
(6.2)

,0 ,
sin(2 / )
AOA AOA AOA m flap
tT
α
αα π φ
=
++
(6.3)
where
X
and
Y
are the coordinates of the center of the flapping wing,
A

is the amplitude of the translational oscillation motion,
f
lap
T
is the flapping
period,
ψ
is the inclined angle of stroke plane.

A
OA
α
is the angle of attack,
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157

,0
A
OA
α
and
,
A
OA m
α
are the angles describing the mean value and amplitude of
the sinusoidal oscillating rotation, respectively.
φ
is the phase difference of
the rotation relative to translation. A schematic diagram is provided in Fig. 6.3
for a better illustration of these parameters. In normal hovering, the wing
strokes in a horizontal plane with
0
ψ
=
while in dragonfly hovering, the
wing strokes in an inclined plane with a non-zero

ψ
. In all our simulations,
the Reynolds number
Re
ref
Uc
ν
=
is defined based on the chord length
c
of
the wing and the velocity
2
ref
flap
A
U
T
π
=
.

6.1.1 Normal hovering mode
Most hovering insects, such as fruit flies, bees and beetles, adopt symmetric
back-and-forth strokes along a horizontal plane (Weis-Fogh 1973), which is
frequently referred to as “normal hovering”. In this subsection, normal
hovering flight is studied.

6.1.1.1 Normal hovering flight without ground effect
The insect-hovering far above the ground is first simulated. Following the

work of Wang et al. (2004) and Eldredge (2007), simulations are carried out at
Re 75=
with characteristic parameters of
2.8A
=
,
,0
/2
AOA
α
π
=

and
,
/4
AOA m
α
π
=
. Three values of
/4
φ
π
=
, 0 and
/4
π
−
for phase

difference are taken into consideration, corresponding to advanced, symmetric
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158

and delayed rotation respectively. For the considered cases, a computational
domain of size 30 30× is capable of providing domain-independent results.
The non-uniform mesh is employed for the domain discretization, with a fine
resolution of 0.02xyh
Δ
=Δ = = in the region involving the insect movement.
Meanwhile, a step size of
0.001tΔ= is used for the time integration.

The drag and lift coefficients
H
C
and
V
C
, which are defined and
normalized in the same way as Eldredge (2007) , are recorded for the first four
flapping cycles in Figs. 6.4 and 6.5. In all the cases, a quasi-periodic state has
been attained after two strokes. A comparison with the published results shows
that for all the three cases, the time histories of
H
C
have good agreements
with both the experimental and numerical results of Wang et al. (2004). The

lift evolution, although it does not match very well with the three-dimensional
experiment of Wang et al. (2004), can capture and well predict the major
features of the force profile, like the timing of the peaks and their values.
Furthermore, it is noted that the agreement between all of our simulation
results and those from Eldredge’s VVPM (2007) is favorably well.

The resultant time-mean drag and lift coefficients
H
C
and
V
C
for the cases
of advanced, symmetric and delayed rotation are (0.598, 0.593), (0.728, 0.464)
and (0.674, 0.262), respectively, where
H
C
and
V
C
are defined as
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159


1
HH
flap

CCdt
T
=
∫
(6.4)

1
VV
flap
CCdt
T
=
∫
. (6.5)
The data reveal that the mean lift drops monotonically with
φ
decreasing
from
/4
π
to
/4
π
−
, indicating that the mean lift is sensitive to the phase
difference, as emphasized by several researchers (Dickinson et al. 1999; Wang
2000; Wang et al. 2004). This variation can be explained from the vorticity
fields (say, in the first half-stroke during the third cycle), as shown in Figs. 6.6
to 6.8 for
/4

φ
π
=
, 0 and
/4
π
−
,
respectively.

After the wing begins a new cycle, a negative LEV and a positive trailing edge
vortex (TEV) are gradually formed and enhanced on the downwind side of the
wing. In the case of
/4
φ
π
=
(Fig. 6.6), the LEV remains attached to the
wing surface throughout the translational process (dynamic stall). This LEV
corresponds to a region of low pressure above the wing and thus will result in
lift enhancement. At the same time, the positive LEV formed in the previous
half-stroke is “recaptured” by the wing (Fig. 6.6(a)-(b)). It combines with the
newly formed TEV and induces the augmentation of rotational circulation,
which in turn contributes to a high lift. For the symmetric rotation case (Fig.
6.7), the LEV attachment is not very significant (comparing Figs. 6.6(a) and
6.7(a)). Although the LEV recapture is also observed (Fig. 6.7(a)), the strength
of the LEV is not as large as that in advanced rotation. Therefore, the
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160

generated lift could not be enhanced remarkably. In the case of delayed
rotation (Fig. 6.8), the wing translates at an angle of attack greater than
/2
π
,
and the flow separates quickly, thus no LEV attachment and recapturing occur.
Meanwhile, a wing with attack angle larger than
/2
π
would result in a
downward lift and all of these lead to a low lift.

6.1.1.2 Normal hovering flight with ground effect
When insects are flying near the ground or perching on bodies, the ground
effect would play an important role. In this subsection, the normal hovering
flight near a surface is simulated. Another important parameter, ground
clearance

c
G
as shown in Fig. 6.3, is introduced to distinguish different flight
conditions. It is defined as the distance between the ground surface and the
wing center (in the time-mean sense). During the assessment of the ground
effect, simulations are carried out at
Re 100
=
. Several ground clearances are
considered while the other parameters are kept fixed at

2.5A
=
,
0
φ
=
,
,0
/2
AOA
α
π
=
and
,
/4
AOA m
α
π
=
. Computational domains of size
30 (15 )
c
G×+ are used to simulate the considered cases. The non-uniform
mesh is used for the domain discretization, with a fine resolution of
0.02xyh
Δ=Δ= = in the region swept by the inset movement. Meanwhile,
0.001tΔ= is selected for the time integration.

The time courses of lift and drag coefficients in one flapping cycle after steady

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
or quasi-steady state has been achieved are depicted at typical clearances
of
1
c
G =
, 3 and 5 in Fig. 6.9. Their comparisons with the corresponding
curves reported by Gao & Lu (2008) demonstrates a satisfactory consistency.
The lift coefficient at
1
c
G =
almost repeats itself in the up- and down-stroke,
and the corresponding drag coefficient, on the other hand, shows
anti-symmetry about each other in the two half-strokes. Furthermore, both the
lift and drag peak around
/0.1
flap
tT
=
and 0.6. This should be explained by
the rotational circulation and wake capture mechanism. For
3
c
G
=

and 5, the
symmetric and anti-symmetric behaviors are lost and the lift generated in the
downstroke is obviously larger than that in the upstroke, peaking
around
/0.8
flap
tT
=
. The drag variation due to the increase in
c
G
from 3 to 5
is not significant except a slight increase in the peak values. The histories of
force coefficients for infinity ground clearance

c
G
=
∞
(corresponding to the
case where the insect is hovering far above the ground) are also included in the
figure. It is observed the two curves almost coincide with those for
5
c
G =
,
which indicates that the ground has little effect on the hovering flight when the
insect flies above the ground at a height no less than
5
c

G
=
.

The time-mean drag and lift coefficients
H
C
and
V
C
have been evaluated
after a steay state of periodicity or quasi-periodicity is achieved and lasts for at
least 10-15 cycles. Their variations versus ground clearance are plotted in Fig.
6.10. To get a clear view of the ground effect on the aerodynamic forces,

H
C

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162

and

V
C
for the flight far away from ground (
c
G

=
∞
), denoted by
,

H
C
∞

and
,

V
C
∞
are also included. It should be noted that the mean drag coefficients
are calculated based on their absolute values, otherwise the drag forces in the
fore- and backstroke would almost cancel out. Fig. 6.10 demonstrates that the
ground has a very significant influence on the aerodynamic forces when the
insect hovers very near the ground (say, at
1
c
G
=
in present study), where
both the lift and drag are greatly enhanced and reach their respective maxima.
As

c
G

increases,

H
C
and
V
C
first decrease quickly to the minima
around
2.5
c
G =
, and then increase asymptotically towards
,

H
C
∞
and
,

V
C
∞
.
In fact, as the insect hovers at a height around
5
c
G
=

above the ground, the
ground effect has become weak enough to be negligible.

The developments of the vortex structure for the first half stroke (forward
stroke) at some representative ground clearances of
1
c
G
=
, 3 and 5 are
depicted in Figs. 6.11-6.13. When the wing begins translation and rotation in
the forward-stroke, a pair of negative LEV and positive TEV is produced.
At
1
c
G =
, the positive LEV formed in the previous back-stroke, due to the
close proximity between the wing and ground surface, flows over the wing to
the downwind side and enhances the newly formed negative LEV (Fig.
6.11(a)-(b)). The LEV is attached to the wing during its translational motion
(dynamic stall mechanism). The TEV, however, is stretched in the narrow gap
and quickly dissipated (Fig. 6.11(b)-(c)). During the stroke reversal, the LEV
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163

formed in the previous half-stroke is separated from the wing (Fig. 6.11(d))
and then sweeps away along the horizontal direction. When
c

G
is increased,
at
3
c
G =
and 5, as shown in Figs. 6.12 and 6.13, the positive LEV formed in
the previous back-stroke combines with the newly produced positive TEV
rather than enhancing the negative LEV (Figs. 6.12(a) and 6.13(a)). At the
same time, the shed vortices at both clearances would not be dissipated as fast
as what happens at
1
c
G =
. After separating from the wing, they move
downwards for quite a long time and then dissipate. At
3
c
G
=
(Fig. 6.12),
the newly generated vortex interacts with the one shed in the previous
half-stroke, forming a complex vortex structure near the ground. At
5
c
G =

(Fig. 6.13), a vortex pair is clearly detected which, as a matter of fact, induces
a downward jet, thus contributing a positive lift on the wing. Therefore, a
higher lift is observed on the wing at

5
c
G
=
than at
3
c
G
=
.

6.1.2 Dragonfly hovering mode
This subsection concerns the flow around a dragonfly hovering wing, which is
frequently observed for some of the best hoverers such as true hoverflies and
dragonflies. The wing is simplified as an ellipse with aspect ratio of 0.25 and
undertakes asymmetric strokes along an inclined stroke plane. For the
convenience of comparison, the kinematic parameters closely follow those in
Wang (2000) and Xu & Wang (2006), where
Re 157
=
,
1.25A =
,
,0
135
AOA
α
=

,

,
45
AOA m
α
=

and
0
φ
=
. A computational domain of size
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164

30 30× is selected for the aerodynamic study under dragonfly hovering mode.
The non-uniform mesh is used for domain discretization, with a locally fine
resolution of 0.02xyh
Δ=Δ= = near the region involving the insect
movement.

The calculations show that the flow can reach a periodic state after only two or
three strokes. The time-dependent drag and lift coefficients during two
consecutive flapping cycles for
75
ψ
=

are depicted in Fig. 6.14 and

compared with the published results (Wang 2000; Gao & Lu 2008; Xu &
Wang 2006; Sudhakar & Vengadesan 2010), from which a satisfactory
agreement can be observed. Fig. 6.14 also shows that in each flapping
cycle,

H
C
has large magnitudes during the stroke reversal process while

V
C

exhibits a peak in the downward stroke, implying that the downstroke has a
greater contribution to the lift generation.

Just as for normal hovering, a pair of LEV and TEV of opposite rotation is
formed in the downward flapping (downstroke) process (Fig. 6.15(a)). The
negative LEV, which is enhanced by interaction (Figs. 6.15(a)-(b)) with the
positive LEV formed in the previous upstroke (wake capture mechanism),
remains attached to the wingtip (Figs. 6.15(a)-(c)) until the stroke reversal.
After the wing reverses its stroke direction and flaps upward, the positive TEV
sheds into the wake (Fig. 6.15(d)). The negative LEV (Fig. 6.15(e)), on the
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165

other hand, combines with a newly formed negative TEV, which is then
coupled with the aforementioned positive TEV, forming a dipole vortex pair
(Fig. 6.15(f)-(h)). This vortex pair then moves downward due to their

respective induced velocities. In particular, a dipole jet is induced by the
vortex pair, generating a lift on the wing.

Simulations are also carried out at three other inclined angles of
30
ψ
=

,
45

and
75

, to give a preliminary view on how the inclined stroke angle
affects the hovering behaviors. The calculated time-mean drag and lift
coefficients in Fig. 6.16 reveal that as the tilt angle of stroke plane increases
from
30

to
75

,

H
C
decreases monotonously while
V
C

increases first and
then decreases, peaking at
60
ψ
=

. The time evolution of force coefficients in
Fig. 6.17 shows that for all the considered cases, while the largest drag
magnitudes appear in the upstroke, their values are reduced as
ψ
increases.
For the lift coefficient, it is seen that two peaks appear in each stroke
for
30
ψ
=

and
45

,
and the second peak which occurs at approximately
/0.3
flap
tT =
is larger. As
ψ
increases to
60


and
75

, only one peak
appears. Despite the discrepancy in peak values and locations, they all exhibit
in the downstroke.

6.2 Particulate flow
Particulate flows widely exist in natural and industrial processes, such as
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
sedimentation of dust particles in aerial, dispersion of fuel particles in
combustion chambers, reaction of catalyst particles in slurry reactors, and
transport of sand particles in rivers, etc. A theoretical analysis of the
underlying physical nature is of fundamental importance. However, these
flows involve not only particle translations and rotations, fluid-particle
interactions, but also some complex physics like particle-particle interaction,
particle-wall interaction, heat and mass transfer, etc., which pose great
challenges and difficulties to numerical methods (Hu et al. 1992; Feng et al.
1994; Hu et al. 2001; Gan et al. 2003; Wang et al. 2008). In this section, these
complex phenomena are tackled with our developed methods. Two particulate
flow problems
−
single elliptical particle motion between closely spaced
walls and cold circular particle settling in an infinitely long channel, are
studied and used to further validate the ability of our developed methods for
simulating particulate flows.

6.2.1 Sedimentation of an elliptical particle between two closely spaced
walls
As the first validation case of particulate flow, the sedimentation of a single
elliptical particle between two closely spaced vertical walls is simulated.
Initially an elliptical particle with major axis
a
and minor axis
b
is
released from zero velocity in a narrow channel of width
L
. The particle (with
density

p
ρ
) is slightly heavier than its surrounding quiescent fluid (with
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
density

ρ
) so it starts to sediment under gravity. Furthermore, the channel is
infinitely long such that the flow behaviors are insensitive to the position
where the particle is released. Due to the strong inertial effects and intense
particle-bounding wall interactions, a series of interesting flow dynamics

would be produced (Hu et al. 1992; Feng et al. 1994; Xia et al. 2009). In the
present simulations, we mainly investigate the wall effect on flow patterns
around an elliptic particle in narrow channels, following the work of Xia et al.
(2009). Under this circumstance, the flow characteristics are described by the
following dimensionless parameters: density ratio
/
ratio p
ρ
ρρ
=
, aspect
ratio
/
A
Rab=
, blockage ratio
/BR L a
=
and terminal Reynolds
number
Re
tmn
tmn
Ua
ν
=
, where

tmn
U

is the terminal velocity of the particle
and

ν
is the kinematic viscosity of the fluid. To save the computational effort,
the ALE technique is introduced into our calculation where the mesh velocity
is set as the vertical velocity of the settling particle. In this way, a fixed
computational domain of size
25aL
×
is used for the simulations where the
top boundary is located
15a downstream and the bottom boundary is located
10a upstream of the particle. A uniform mesh with resolution /52ha= and
time step size
0.0001t
Δ
= is employed for all the following simulations.

Since our simulations target on narrow channels, the particle will inevitably
wander close to the walls of the narrow channel under certain situations. To
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
avoid the penetration of particles into the wall and the breakdown of numerical
calculations, a particle-wall collision model is employed, which illustrates that
an artificial repulsive force should be introduced when the gap between the
particle and the wall is less than some threshold distance. A spring force

formula which was originally suggested for circular particles by Glowinski et
al. (2001) and later revised specially for elliptical particles by Xia et al. (2009)
is utilized, which is given as

()
()
2
2
0,
1
/,
c
repulsive
cc c c
w
C
ζ
ζζ
εζ
⎧−>
⎪
=
⎨
⎛⎞
−
−−− −≤
⎪
⎜⎟
⎝⎠
⎩

xX
F
xX X x xX xX

(6.6)
In the above formula,
c
X
is the point closest to the wall on the elliptical
particle,
x
is the corresponding point on the wall, so
c
−xX
gives the
minimum distance between the wall and the points on the particle.
C
is a
scale for the relative gravitational force defined as
(
)

p
p
CAg
ρρ
=−
,
where

p
A
ab
π
=
is the area of the particle.
w
ε
is a stiffness parameter and
ζ
is the aforementioned threshold distance. Both parameters are to be
determined empirically.

According to the work of Xia et al. (2009), five
BR
regimes are roughly
established depending on the sedimentation patterns of the elliptical particle:
Regime A (
1.0BR ≤
): the particle oscillates around the centerline while
wiggling down the channel;
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
Regime B (
1.0 1.5BR<<
): the magnitude of the particle oscillation
increases which eventually gives way to tumbling where the particle rolls

down along one of the walls of the channel;
Regime C (
1.5BR ∼
): the particle achieves a steady-state vertical settling
(major axis parallel to the centerline of the channel) for a small range of
blockage ratios;
Regime D (
1.5 2.0BR
<
≤
): the particle migrates away from the centerline and
sediments with a constant inclination to the horizontal;
Regime E (
2.0BR >
): the particle translates to the centerline once again and
settles steadily with a horizontal orientation (minor axis parallel to the
centerline).

Taking the same parameter combination as that in Xia et al. (2009):
0.1acm=
,
21
0.01cm s
ν
−
=
,
1.1
ratio
ρ

=
and
2
A
R
=
, our simulations produce
a group of results at different blockage ratios.

The snapshots of the particle’s position after steady sedimentation has reached
are displayed in Fig. 6.18 at blockage ratios of
12 /13BR
=
, 18/13, 20/13,
22/13 and 32/13, from which their respective sedimentation modes are clearly
visualized. Together with the corresponding trajectories plotted in Fig. 6.19,
five sedimentation patterns: oscillating, tumbling, vertical sedimentation,
inclined sedimentation and horizontal sedimentation are clearly identified in
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
the order of increasing blockage ratio, which have well reproduced the mode
transitions observed by Xia et al. (2009). The instantaneous vorticity fields
developed at different blockage ratios are also shown in Fig. 6.20. As expected,
the vorticity fields are symmetric about the centerline at vertical and
horizontal sedimentation modes while under the other circumstances, the
vortex always shows a larger size on the opposite side toward which the
particle deviates.

6.2.2 Cold particle settling in an infinitely long channel
A cold particle with uniform and constant temperature sedimenting in an
infinitely long channel is a mixed convection problem. In such a thermal
particulate flow, both natural and forced convections are involved and their
interaction and competition would produce a number of interesting flow
behaviors. This problem was first studied by Gan et al. (1994) via a
finite-element-based ALE (Arbitrary Lagrangian Eulerian) method. They
presented a detailed analysis on the numerous flow features and the underlying
mechanisms. Later, Yu et al. (2006) revisited this problem as a validation case
to test their fictitious domain method. Similarly, Feng & Michaelides (2008),
during their numerical investigation on particulate flows with thermal
convection, also utilized this case to verify their code.

Following Gan et al. (1994) and Yu et al. (2006), a circular particle of
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diameter

p
d
and density

p
ρ
is released with zero initial velocity in an
infinitely long vertical channel. The channel has a width

4
p
d
and contains a
hot fluid with uniform temperature
T
∞
and density
ρ
. The particle sediments
under gravity and maintains its surface temperature as constant and fixed
at

p
T
(
<T
∞
) during the sedimentation. From the viewpoint of computational
efficiency, the ALE technique is employed in our calculations where the mesh
velocity is set as the vertical velocity of the sediment particle. In this way, a
fixed computational domain of size
32 4
pp
dd
×
is selected where the top
boundary is located
24
p

d downstream and the bottom boundary is located
8
p
d upstream of the particle. A uniform mesh with resolution /32
p
hd=
and time step size
0.005tΔ= is used for all the considered simulations.
Choosing appropriate reference quantities, the problem is characterized by
dimensionless parameters: density ratio
/
ratio p
ρ
ρρ
=
, Prandtl
number
Pr
p
c
k
μ
=
, Grashof number
(
)
23
2
pp
TTdg

Gr
ρβ
μ
∞
−
=
and Reynolds
number
Re
ref p
Ud
ρ
μ
∞
=

where the reference velocity is defined as
(1)
ref ratio
Ur g
πρ
=−
. In the present study, the characteristic parameters are
set at the same values of
1.00232
ratio
ρ
=
and
Pr 0.7

=
as in Gan et al. (1994)
and Yu et al. (2006). The specific value of
Re
is not given in the work of
Gan et al. (1994). However, the terminal velocity based Reynolds number
*
Re Re
tmn tmn
U=
reported by them for the isothermal case (
0Gr =
) is 21,
which is yielded by setting
Re 40.5
=
in our tests (actually, our simulation
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
with the aforementioned parameters produces
Re 20.96
tmn
=
). So all our
simulations below are carried out at
Re 40.5
=

.

Gan et al. (1994) identified that the particle lateral equilibrium positions as
well as the wake structures and temperature fields behind the cold settling
particle exhibit great variantion for different Grashof numbers, which are
classified by them into five
Gr
regimes:
Regime A (
5000 << Gr
): the particle settles steadily along the centerline, and
the vorticity as well as temperature field is steady and symmetric, with a pair
of vortices standing behind the particle.
Regime B (
810500 << Gr
): onset of vortex shedding appears behind the
particle. The particle migrates slightly away from the centerline and develops
a small-amplitude lateral oscillation.
Regime C (
2150810 << Gr
): the particle reaches a steady sedimentation
close to one of the walls and the vortex shedding is suppressed.
Regime D (
45002150 << Gr
): the particle sediments steadily and the
centerline once again becomes the stable equilibrium position. Vortex
shedding still remains absent and a pair of long vortices stands behind the
particle.
Regime E (
4500>Gr

): the flow in this regime is characterized by the
re-emergence of lateral oscillations about the centerline.

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
In the following, our simulation results for different
Gr
are presented and
compared with Gan et al. (1994) and Yu et al. (2006). Fig. 6.21 plots the
streamlines (in a reference frame fixed on the particle) together with the
vorticity and temperature contours at
100Gr
=
, 564, 1500, 2500 and 5000,
while Fig. 6.22 presents the corresponding time-dependent lateral particle
positions, from which the flow features of all five
Gr
regimes reported by
Gan et al. (1994) and Yu et al. (2006) are observed. In particular, Gan et al.
(1994) recognized the existence of two migrating types in regime C: the
natural extension of Regime B for lower
Gr
in which the particle oscillation
slowly dies out while gradually migrating away from the centerline and finally
gives way to steady sedimentation; and the simple migration off the centerline
for higher
Gr

in which the particle oscillation never appears. Both behaviors
(see
1000Gr =
and 2000 in Fig. 6.22) are well reproduced in the present
simulation. The equilibrium position of the particle center at
1400Gr =
is
roughly midway between the channel centerline and the wall, which matches
well with the results of Gan et al. (1994).

It is also noted that our results at high
Gr
have a good agreement with Gan et
al. (1994) but differs significantly from Yu et al. (2006). For example,
at
4500Gr =
, Yu et al. (2006) observed a turbulent-like flow in which the
particle oscillates violently and regularly, while such a feature was not
detected either in the present study (Fig. 6.21(e)) or in Gan et al. (1994). Fig.
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174

6.23 presents the variation of terminal-settling-velocity based Reynolds
number
Re
tmn
versus the Grashof number. Their comparisons with those
provided by Gan et al. (1994) and Yu et al. (2006) show a fairly nice

agreement.

6.3 Forced Convective Heat Transfer from a Transverse
Oscillating Cylinder in the Tandem Cylinder System
The investigation of flow and forced convection around multiple circular
cylinders is of significant importance in fluid mechanics and heat transfer.
Literature review shows that the fluid field and thermal performance from a
circular cylinder are very sensitive to either its transverse oscillation or the
upstream disturbance (Bishop & Hassan 1964; Zdravkovich 1982; Ongoren &
Rockwell 1988; Gau et al. 1999; Park & Gharib 2001; Pottebaum & Gharib
2006; Igarashi 1981; Summer 2010; Kostic & Oka 1972; Mahir & Altac 2008;
Han et al. 2010). Thus, it is reasonable to expect that the combined effect of
the cylinder vibration and the presence of an upstream cylinder may produce
even more complex influence. On the other hand, it should be indicated that so
far, there is no report on heat transfer analysis concerning their mutual effect.
This motivates us to examine the behaviors of thermal flow around two
tandem cylinders when the downstream cylinder is forced to oscillate
transversely. As a first approximation to the problem, a two-dimensional
simulation at low Reynolds number is used to give some insight into the
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
dependence of the wake, drag, lift coefficients and Nusselt number on the
cylinder spacing, vibration frequency and amplitude.

The configuration of the flow system is shown in Fig. 6.24. Two isothermal
cylinders of equal diameter are arranged in tandem, with a center-to-center
separation spacing of

G

(normalized by
D
). The upstream cylinder is
stationary and exposed to a cold free stream. The downstream cylinder
undergoes a controlled transverse oscillation of the form

sin(2 )
c
yA ft
π
=
, (6.7)
where
A
is the oscillation amplitude (normalized by
D
),
c
f
is the
oscillation frequency. As a result, the fluid and thermal flow behaviors are
characterized by five most important parameters,
Re, Pr, ,
A
c
f

and

G
. In
the present study, we focus on the influence of
,G
A
and

c
f
on flow
behaviors by fixing
Re 100=
and
Pr 0.7
=
. The computational domain is of
size
45 30×
, which has shown to be sufficiently large for the present problem.
The top and bottom boundaries are with the condition of
0
T
yy
∂
∂
=
=
∂∂
u
. At the

inlet which is located
12 upstream of the cylinder, velocity
)0,1(=u
and
temperature
0
=
T
are specified. At the outlet, the Neumann condition of
0
T
x
x
∂∂
==
∂∂
u
is applied. On the surface of two cylinders, the constant
temperature
1
c
T
=
and no-slip boundary condition
0
=
u

are imposed. The
grid independence study shows that a grid with a uniform mesh size of

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
0.02xyΔ=Δ=
around the cylinders and a time step size of
0.005tΔ=
is
fine enough to provide accurate results for
Re 100
=
and
Pr 0.7
=
.

According to our observation as well as that of Yang & Zheng (2010), three
flow regimes have been identified for the tandem-cylinder system based on
whether the shear layer is separated from the upstream cylinder. They are
respectively vortex shedding (“VS”) regime, critical regime and vortex
formation (“VF”) regime. Among them, the critical regime exists for a narrow
spacing range as a transition zone separating the two completely distinct “VS”
regime and “VF” regime. Three spacings of
2, 4G
=
and
7
are selected as
their representatives to illustrate the frequency and amplitude effect of

cylinder vibration on the flow and thermal patterns in each flow regime. In
each spacing, two small amplitudes of
0.15A
=
and 0.35 and eight
excitation frequencies are examined. The eight frequencies are the natural
shedding frequency (
s
t
f
is corresponding to the two tandem stationary
cylinders), two proximate frequencies (
0.9
s
t
f
,
1.1
s
t
f
), two lower (
0.4
s
t
f
,
0.7
s
t

f
)
and three higher (
1.3
s
t
f
,
1.5
s
t
f
,
1.7
s
t
f
) exciting frequencies.

6.3.1 Vortex structure
Based on our calculation, the response of the near-wake structure can be
classified into two basic categories. In the first category, the vortex formation
is phase-locked with respect to the motion of downstream cylinder, whereby
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
the near-wake structure is highly repetitive from cycle to cycle of the cylinder
oscillation. However, this phase-locked behavior of vortices is found to be

dependent upon the gap spacing and excitation parameter of the downstream
cylinder. The other category refers to a modulated structure, in which the near
wake changes its pattern from one cycle of cylinder oscillation to another
cycle.


In the analysis below, a general response characteristic of the cylinder system
is firstly described. For the phase-locked case, the variations on the near-wake
geometry are also examined. To facilitate direct comparison, without special
illustration, all the visualizations are depicted at the instant when the
downstream cylinder is at its maximum positive position.

6.3.1.1 In the “VS” regime,
2G
=

For
2G =
, the flow between the two tandem cylinders is in the “VS regime”:
there is no vortex shedding in the gap region under the present oscillations.
The pair of shear layers separated from the upstream cylinder reattaches to the
downstream one. The instantaneous wake behind the downstream cylinder, on
the other hand, is largely influenced by the excitation frequency. We first look
at the cases of
0.15A=
. At low oscillating frequencies ranging from
/0.4
cst
ff=
to 1.3, the vortex pattern (Fig. 6.25(a)) corresponds to the

well-known 2S mode as indicated by Willianson & Roshko (1988), in which