Development of immersed boundary methods for isothermal and thermal flows 8

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Chapter 8

Applications of Developed IBM Solvers to Simulate
Three Dimensional Moving Boundary Flows

In this chapter, the new boundary condition-enforced IBM proposed in the
framework of NS solver is utilized to study the three-dimensional moving
boundary flows where the immersed objects are undertaking complex and
prescribed motions. Two biomimic problems are considered. The first one
discusses the flow behaviors around a heaving and pitching finite span foil and
the second one investigates the hydrodynamic performances around a fish-like
swimming body.

8.1 Incompressible flow over a heaving and pitching finite span
foil
The three-dimensional incompressible flow around a finite span foil under a
heaving and pitching motion has been investigated by several researchers. Von
Ellenrieder et al. (2003) conducted a visualization study on the
three-dimensional flow behaviors behind a rectangular heaving-and-pitching
foil of aspect ratio (span-to-chord) 3.0 at a Reynolds number of 164 and found
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
sets of rings- and loops-like structures. A range of Strouhal numbers (between
0.2 and 0.4), pitch amplitudes (between
0
D
and
20
D
) and heave/pitch phase
angles (between
60
D
and
120
D
) were tested which showed that the variation
of these parameters had visible effects on the wake structures. Blondeaux et al.
(2005), by employing a panel method, numerically investigated the vortex
topologies behind the same foil as in the experiment of von Ellenrieder et al.
(2003). Based on the numerical calculations, they identified that the vortex
rings produced by the foils were shed every half a cycle. However, the vortex
topologies observed in their numerical work were not completely consistent
with those in the visualization of von Ellenrieder et al. (2003). Buchholz et al.
(2006) performed flow visualization for the interrogation of wake structures
generated by a rigid flat pitching panel of aspect ratio 0.54, from which they
observed horseshoe vortices of alternating sign shed twice per flapping cycle.
They also identified the robustness of the wake patterns with respect to
changes in Reynolds number, aspect ratio and amplitude. Dong et al. (2006)
carried out a detailed numerical analysis on the hydrodynamic mechanisms
associated with the thrust generation of thin ellipsoidal flapping foils. The

thrust and propulsive efficiency of these foils were also examined over a range
of span-chord-ratios and Strouhal numbers as well as pitch-bias angles. They
indicated a monotonic increase of thrust coefficient with aspect-ratio and
Strouhal number for all foils. Shao et al. (2010) numerically predicted the
wake structures and propulsion performance of a finite-span flapping foil with
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
five different aspect ratios at
Re 200
=
and reported that both the thrust
coefficient and propulsion efficiency increase with increasing aspect ratio.

In this subsection, the wake structure and propulsion performance of rigid
finite-span foils heaving and pitching in a free stream is studied. The geometry
of the foil is shown in Fig. 8.1. It has a rectangular platform and a symmetric
cross section whose profile is similar to a NACA0012 profile. The chord and
span width are denoted by
c
and
W
, while the area of the platform is
denoted by
S
. The aspect ratio of the foil is defined as:

22

area of the rectangular platform
(length of the chord)
SW
AR
cc
===
(8.1)
The foil is placed in a free stream with uniform incoming velocity
U
∞
, and
oriented with
x
-axis along the streamwise direction and
y
-axis along the
spanwise direction (Fig. 8.1). In the current simulations, the foil undergoes a
combination of harmonic pitching (angular oscillation) and heaving (vertical
oscillation) motion where the foil pitches about its mass center according to

0
sin(2 )
c
ft
ϑ
ϑπψ
=+
, (8.2)
while the foil center, at the same time, heaves in the
z

-direction according to

0
sin(2 )
c
hh ft
π
=
. (8.3)
In Eqs. (8.2) and (8.3),
0
ϑ
and
0
h
are pitch and heave amplitudes, and

ψ

is the phase difference between the pitching and heaving motion.
c
f
is the
oscillating frequency. To better illustrate the combined “pitching-and-heaving”
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
motion, a schematic diagram is appended in Fig. 8.2 in its side view.

By taking the chord length
c
and free stream velocity
U
∞
as the reference
length and velocity scales respectively, the problem can be characterized by
several other key dimensionless parameters in addition to
A
R
,
0
ϑ
and

ψ
:
the Reynolds number
Re
Uc
ν
∞
=
, the Strouhal number
0
2

c
hf

St
U
∞
=
and
normalized heave amplitude
0
/hc
. The present study focuses on the variations
of the wake structures and propulsion performance with respect to the aspect
ratio

A
R
and Strouhal number
St
at Reynolds number
Re 200=
. The
variation range of

A
R
and
St
is summarized in Table 8.1. Other involved
parameters are set as
0
/0.5hc
=

,
0
30
ϑ
=
D
and
90
ψ
=
D
. A computational
domain of size 15 10 10ccc×× is chosen for the simulations, where the mass
center of the foil is initially located at (
5c ,5c ,5c ). As always, non-uniform
meshes are employed for domain discretization, with a fine resolution of
/40xyzhcΔ=Δ=Δ= = covering the region swept by the flapping motion of
the foil. For the time integration, a step size of
4
510t
−
Δ= × is used.

To have an idea of the wake evolution and vortex shedding process of this
rectangular flapping foil, the vortex structures in their
2
λ
-definition at four
phases in one flapping cycle is plotted in Fig. 8.3 for
3AR

=
and
0.6St =
.
The left column gives the perspective views and the right column gives the
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280

side view. In the first half cycle, the foil is first moving upward and then back
to the equilibrium position. During its upward motion, it is observed that the
flow separates from the leading edge along the lower foil surface (Fig. 8.3(a)).
The free shear layer along the upper foil surface, on the other hand, rolls up
and forms a negative vortex structure in the clockwise rotation (Fig. 8.3(b)).
When the foil reverses its motion, this negative vortex structure is shedding
from the trailing edge and a new negative vorticity is generated along the
upper surface (Fig. 8.3(c)). At the same time, the shed vortex connects with
the two tip vortices from the span ends, forming a vortex ring or vortex loop.
In the second half cycle, the foil first heaves downward to the lowest position
(Fig. 8.3 (d)) and then back to the central position again (Fig. 8.3(e)). The
evolution of vortex structures in this half cycle follows a mirror image of those
in the first half cycle. Thereafter, the downstream wake of this foil consists of
two sets of vortex rings. These observations are consistent with the findings
reported by Dong et al. (2006) for a thin ellipsoidal flapping foil and Shao et al.
(2010) for the same rectangular foil.

A full visualization is shown in Fig. 8.4 by plotting a top view of the vortex
structure at the instant corresponding to Fig. 8.3(d). As the vortex rings are
convected downstream, a narrowing of their spanwise lengths is observed, a

phenomenon which has also been noted in the simulations of Blondeaux et al.
(2005) for a similar rectangular flapping foil and in the experiments of
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281

Buchholz & Smits (2006) for a pitching plate. Dong et al. (2006) has also
noticed this feature for their ellipsoidal foil.

The general fluid dynamic behaviors for this rectangular foil (taking
3AR =

and
0.6St =
as a representative) are also viewed by examining contours of
the spanwise vorticity and contours of the mean streamwise velocity along the
spanwise symmetry plane, which are respectively presented in Fig. 8.5 and Fig.
8.6. In Fig. 8.5, the formation of an inverse Karman vortex street is clearly
shown, which, as discussed earlier, is always associated with the
thrust-production. Contours of the mean streamwise velocity in Fig. 8.6, where
the mean value was calculated over several consecutive flapping cycles after
the steady force behaviors have been reached, show the existence of a single
streamwise directed jet immediately behind the trailing edge of the foil. This
high-intensity jet develops into a bifurcated one about three chord-lengths
downstream of the trailing edge in the wake.

The most significant wake behavior produced by the finite-span rectangular
flapping foil has been discussed. Based on this knowledge, the effect of aspect
ratio and Strouhal number is examined in the following.

The aspect ratio analysis is limited to
0.6St
=
. The vortex structures at
different aspect ratios (
1, 2, 4AR
=
and
5
) are given in Fig. 8.7 in
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282

perspective and top visualizations. By comparing them with the case
of
3AR =
in Fig. 8.3, it is noted that at small
A
R
, i.e.
1
A
R
=
, 2 and 3, the
wakes are dominated by two sets of vortex rings. The two sets of vortex rings
separate from each other in the wake and convect downstream at an oblique

angle to the wake centerline. Furthermore, the smaller the aspect ratio, the
more circular the vortex rings and the larger the oblique angle. At large

A
R
,
i.e.
4
A
R =
and 5, due to the large span size, the tip vortices are not
strengthened enough to merge together and the wake is dominated by two sets
of vortex loops. These vortex loops, as can be clearly observed in Fig. 8.7(c)
and 8.7d), wrap end-to-end with their neighbors and keep twined together in
the wake. Despite of the differences, the wake structures share a number of
interesting features. For all the cases, the vortex rings/loops contract
themselves in the spanwise direction as they move downstream. However, the
streamwise length of the vortex rings/loops for different
A
R
remains nearly
the same. Moreover, the vortices seem to shed from these flapping foils almost
at the same frequency, by noticing the same number of vortex rings in each
case.

Fig. 8.8 presents contours of the mean streamwise velocity in the spanwise
symmetry plane (
0y =
) for
1, 2, 4AR

=
and
5
. Just as the corresponding
plot for
3AR =
, the wake in each case induces a jet behind the flapping foil.
However, the topologies of the wake are quite different from each other. For
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283

cases of small span size of
1
A
R
=
and 2, the jet bifurcates itself downstream
of the trailing edge so that the wakes in Figs. 8.8(a) and 8.8(b) show a
bifurcated shape. A closer examination shows that the bifurcation location
moves downwards as the span size of the flapping foil increases. For example,
at
1
A
R =
, the jet bifurcates at about one chord-length downstream of the
trailing edge while at
2
A

R
=
the bifurcation happens around two
chord-lengths away. Compared with the case of
3AR
=
shown in Fig. 8.6, the
bifurcations in the previous two low-
A
R
cases appear earlier and their
bifurcated patterns are clearer as well. In Figs. 8.8(c) and 8.8(d), no clear
bifurcation is observed and a single jet maintains up to about two or three
chord-lengths behind the trailing edge, showing that a quite distinct wake
pattern is produced for high aspect ratios of
4
A
R
=
and 5. A roughly
quantitative analysis reveals that the strength of the jet increases with an
increase in the span size, which could be fairly consistent with the above
qualitative discussions.

To evaluate the effect of Strouhal number, simulations at three other values of
0.35, 0.8St =
and 1.0 are carried out for
3AR
=
. The top views of the

vortex structures for these cases are presented in Fig. 8.9. In combination with
the plot in Fig. 8.4 for
0.6St
=
, it is noted that the wake at the
low-flapping-frequency case of
0.3St
=
is rather different from the other
three. At
0.3St =
, as shown in Fig. 8.9(a), no linkage exists between the tip
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284

vortices so that the vortices appear as loops rather than rings, which could be
attributed to the lower strength of the tip vortices at this low flapping
frequency. With low strength, the interaction between the tip vortices is
weakened and they are able to be combined together and form a ring. The
three larger-
St
cases in Fig. 8.4 and Figs. 8.9(b)-(c) share similar vortex
structures. The two tip vortices are linked together and the vortex rings are
clearly displayed. Moreover, it is observed that as the Strouhal number
increases, the spanwise length of the vortices is reduced (for clarity, consider
comparison of the spanwise spacing between the second pair tip vortices in
each figure). This is because the tip vortices have different vortex strengths at
different flapping frequencies. At higher ones like

1.0St
=
, they are capable
of moving closer together under their mutual induction.

As previously, contours of the mean streamwise velocity in plane
0y =
are
plotted. At the low frequency case of
0.3St
=
in Fig. 8.10(a), a single jet
region with relatively low streamwise velocity appears immediately behind the
trailing edge, which does not seem to bifurcate in the downstream wake. At
higher frequency cases of
0.8St
=
and 1.0 in Figs. 8.10(b) and 8.10(c), a
diamond-shaped region of high velocity exists immediately behind the
flapping foil and there is no evidence of clear bifurcation for this high-velocity
jet. However, unlike the low frequency case, a large region of high streamwise
velocity reappears around the centerline in the downstream wake region.
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
Finally, the hydrodynamic performance of this rectangular flapping foil is
assessed in terms of thrust coefficient
T

C
and propulsion efficiency

η
, which
are, respectively, defined as

2
1
2
T
T
F
C
US
ρ
∞
=
(8.4)

T
FU
P
η
=
(8.5)
where

T
F

and

T
F
are the instantaneous and time-mean thrust, respectively,
while
P
is the time-mean input power.

Fig. 8.11 shows the variation of time-mean thrust coefficient versus foil
aspect-ratio for
0.6St =
. It is found that the thrust increases monotonically
with the aspect ratio and might approach a constant with a further increase
in

A
R
. Such a behavior is recognized in the simulation of thin ellipsoidal
flapping foils by Dong et al. (2006) and well documented for the
two-dimensional flapping foils by Anderson et al. (1998) and Jones et al.
(1998). The numerical result of Shao et al. (2010) is also included for a direct
comparison, from which a good agreement is obtained except the present
thrust coefficient is a litter larger than the result of Shao et al. (2010) at
1
A
R =
.
Similar trend also holds for the propulsive efficiency in Fig. 8.12, where

η

increases monotonically with respect to
A
R
and asymptotically approaches a
constant value.
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286

The variations of the propulsive efficiency and time-mean thrust coefficient
with respect to Strouhal number at the intermediate aspect ratio
3AR =
are
plotted in Fig. 8.13. While the thrust shows a monotonic increase with the
Strouhal number, it is so small at
0.3St
=
and almost invisible. The desirable
thrust can only be achieved at a relatively larger Strouhal number. However, it
does not mean that a larger Strouhal number will always give a better
efficiency. The efficiency variation implies that the efficiency achieves a
maximum at an intermediate Strouhal number and with a further increase in
Strouhal number, the efficiency would decrease instead.

8.2 Hydrodynamics of flow over a fish-like body in carangiform
swimming
Fish offers an attractive self-propulsion paradigm through a relatively simple

but remarkably robust undulatory locomotion, in which the fishes usually
propel themselves forward by generating and propagating waves backwards
along their flexible bodies (Lighthill, 1969). Experimental observations show
that the majority of fish families use body/caudal fin (BCF) as their routine
propulsor (Videler 1993; Borazjani & Sotiropoulos, 2008). Within the BCF
propulsion, two general propulsive modes can be broadly distinguished
(Lindsey, 1978): carangiform swimming and anguiliform swimming. When
the undulation wave is characterized as a large and almost constant amplitude
along the whole body, the swimming is said to be in anguiliform mode. On the
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
other hand, if the large amplitude of the propulsive wave is mostly confined to
the posterior part of the body and increases significantly in the caudal area, the
swimming mode is called carangiform.

Recent experiments and computations have shed valuable insights into the
performance and hydrodynamic details of fish-like swimming. With the help
of the highly advanced PIV technique, knowledge on the swimming
kinematics and induced wake fields has been obtained (Muller et al. 1997;
Drucker & Lauder, 2002; Nauen & Lauder 2002; Tytell & Lauder, 2004).
Resorting to the laboratory flexible fish-like mechanical models, repeated
accurate measurements of the hydrodynamic forces on a fish-like swimming
body is allowed (Triantafyllou & Triantafyllou, 1995). As a useful and even
indispensable auxiliary mean on investigating this subject, numerical
simulations present an increasing popularity despite the considerable challenge
related to the rapid change of underlying fish geometry. The early numerical
studies, such as those of Barrett et al. (1999), Wolfgang et al. (1999) and Zhu

et al. (2002), were based on potential flow theory, i.e., the assumption of
inviscid flow, due to the inertial feature of flow regimes they considered. By
developing an inviscid boundary integral method, Barrett et al. (1999)
uncovered a plausible mechanism responsible for the drag reduction in
fish-like locomotion. By employing an inviscid panel method, Wolfgang et al.
(1999) and Zhu et al. (2002) made very comprehensive studies on the wake
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
structures of a steady straight-swimming giant danio. To enable an accurate
calculation of the drag force experienced by the fish, the viscous effect was
taken into consideration afterwards. Three-dimensional viscous simulations on
tadpole swimming have been reported by Liu’s research group (Liu et al. 1996;
Liu & Wassersug, 1997; Liu & Kawachi 1999), from which, the dynamic
behaviors and propulsion mechanisms related to this unusual vertebrates were
generally understood. Three-dimensional viscous studies on a self-propelled
body in anguiliform swimming have been carefully analyzed by Kern &
Koumoutsakos (2006) and later revisited by Zhou & Shu (2012) using their
local domain-free-discretization (DFD) method. Borazjani & Sotiropoulos
(2008, 2009), via a hybrid Cartesian/immersed boundary method,
systematically investigated the hydrodynamics of carangiform as well as
anguiliform swimming spanning transitional and inertial flow regimes.

In this section, the carangiform swimming which is used by large numbers of
fishes such as mackerel Scomber Scombrus is considered. An anatomically
realistic model of a mackerel fish provided by Borazjani & Sotiropoulos (2008)
is employed as the flexible fish-like object. It is constructed by slicing the real
fish along its body axis in several fillets and carefully measuring the

cross-sections of each fillet and then stacking them together. All fins except
the caudal fin are neglected in the model, as shown in Fig. 8.14. The fish
swimming is assumed to be steadily along a straight line at constant speed
U
∞
,
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
whose specific undulation kinematics is prescribed based on the experimental
observations of Bideler & Hess (1984) by specifying the lateral displacement
of the fish backbone as a function of time. It is expressed in a
backwards-travelling wave with nonlinear-varying amplitude as follows:

(/,) (/)sin( / )hx Lt Ax L kx L t
ϖ
=−
, (8.6)
where

x
 is in the axial (swimming) direction, measured along the fish axis
from the tip of the fish head,

L
 is the fish length,
(/)Ax L
is the nonlinear

amplitude envelope of lateral undulation along the fish body, and
(/,)hx Lt
is
the lateral excursion at time
t
.
ϖ
denotes the angular frequency of the body
undulations while

k
represents the wave number and is correlated with the
wavelength

λ
in
2
k
π
λ
=
. As a result, the fish-like swimming is characterized
by four dimensionless similarity parameters: two dynamic ones plus two shape
parameters. The two dynamic parameters are: (1) the Reynolds
number
Re
UL
ν
∞
=

, based on the fish length
L
, swimming speed
U
and
fluid kinematic viscosity
ν
, and (2) the tail beat Strouhal
number
max
2/St fh U=
, based on the maximum lateral excursion of the
tail
max
2
A
h=
, and the tail beat frequency
f
. The two shape parameters are: (1)
the dimensionless wavelength
/
L
λ
, and (2) the dimensionless undulation
amplitude envelope
(/)/Ax L L
, which in the present work as well as in
Borazjani & Sotiropoulos (2008), is described by a quadratic function
()

2
01 2
(/) / /Ax L a ax L a x L=+ +
 with
0
0.02a
=
,
1
0.08a
=
−
and
2
0.16a =
,
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
giving the maximum lateral excursion
max
0.1h
=
at the fish tail. A
computational domain of size
82LL L
×
× is chosen for the simulations,

while the fish head is initially located at (
2L , 0.5L , L ). As always, a
non-uniform mesh is employed for domain disretization, with a very fine
resolution of /100xyzhLΔ=Δ=Δ= = in the local region near the
swimming fish. For the time integration, a step size of
0.01t
Δ
= is chosen.

The present work concerns both the transitional and inertial flow regimes, so
simulations at two Reynolds numbers of
Re 300
=
and 4000 are carried out
as in the work of Borazjani & Sotiropoulos (2008). In each flow regime, a
range of systematically varying Strouhal numbers is simulated and the shape
parameters are kept fixed throughout our simulations with the wavelength
taken to be
/0.95L
λ
=
.

The three-dimensional flow structures induced by the swimming fish are
visualized in Figs. 8.15 and 8.16, where the instantaneous vortex structures in
their
2
λ
-definition are depicted, respectively, for
Re=300

and 4000 at some
representative
St
. Our results reveal two different wake patterns at each
Re
.
Based on the layout of the generated vortex structures, the single-row wake
and double-row wake as reported in the work of Borazjani & Sotiropoulos
(2008) are well observed. In Figs. 8.15(a) and 8.16(a) which correspond,
respectively, to
0.3St =
at
Re=300
and
0.2St
=
at
Re=4000
, the single
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291

row vortex structure is presented. This single-row wake, as shown, is confined
within a relatively narrow strip centralized along the fish body axis. In Figs.
8.15(b) and 8.16(b) where the
St
is higher, the wake goes through a lateral
divergence and the vortices spread away from the centerline of the body,

forming two rows of vortices in a wedge-like arrangement. Our simulations
also show that the wake patterns depend on both
Re
and
St
but primarily
on
St
. The exact value of
St
at which the wake pattern transits from the
single row to double row is
Re
-dependent. The instantaneous in-plane
streamlines and vorticity field in the mid plane of
0y
=
at
0.3St =
are
depicted for both
Re
in Fig. 8.17. As expected, at a lower
Re
where viscous
effects play a dominant role, the viscous region around the fish body is thicker
and the overall width of the wake is larger.

The time-dependent hydrodynamic forces experienced by the fish
for

Re 300=
and 4000 are plotted in Figs. 8.18(a) and 8.18(b), respectively,
in forms of force coefficient
F
C
which is defined as

22
/
Fx
CFUL
ρ
∞
=
, (8.7)
where

x
F
is the hydrodynamic force in the
x
-direction. Note that the
hydrodynamic force

x
F
is different from the drag force
D
F
. Furthermore, all

the values of
F
C
presented herein and what follows are normalized by the
calculated force coefficient
,0

F
C
for the rigid body fish (i.e.
0St =
) at the
same
Re
. If
0
F
C >
, the net force exerted on the fish is consistent with its
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
swimming direction, thus contributing positively to the fish swimming.
Thereafter, the net force under this situation is referred to as being of thrust
type. On the other hand, the net force with negative value
0
F
C <

basically
goes against the fish swimming and thus referred to as being of drag type.
The time-evolution curves in Figs. 8.18(a) and 8.18(b) show several
remarkable features of the hydrodynamic force caused by the fish undulation:
(1)
For all the
St
under consideration, the force coefficients exhibit two
peaks in each tail-stroke cycle: one during forward stroke and the other
during backward stroke, which is consistent with the experimental
discoveries of Hess & Videler (1984) and numerical observations of
Borazjani & Sotiropoulos (2008);
(2)
As
St
increases from the rigid case of
0St
=
, the force coefficient
appears to be of drag-type throughout the entire stroke cycle until a
critical
St
(
1.0St ≈
for
Re 300
=
and
0.3St
≈

for
Re 4000
=
), where
the thrust-type regime begins to appear. With
St
increasing further
beyond the critical
St
, the thrust-type regime in each stroke cycle
becomes wider and larger.
(3)
At small
St
, approximately
0.3St
<
, the net force is of drag-type and with
its magnitude greater than the corresponding value of the rigid case of
0St =
at the same
Re
. At moderate
St
(
0.3 0.5St
<
<
for
Re 300=

and
0.5St ≈
for
Re 4000=
), although the force still remains of drag-type,
its magnitude lowers down as compared to its rigid counterpart.
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
These hydrodynamic features hold essentially in both transitional and inertial
flow regimes. The critical
St
, however, depends on the
Re
.

The time-mean force coefficient
F
C
, which is calculated by averaging the
force coefficient

F
C
over several consecutive swimming cycles, is presented
in Fig. 8.19 as a function of
St

. It is observed that for
Re 300
=
, our results
match perfectly well with those of Borazjani & Sotiropoulos (2008) and Zhou
& Shu (2012), and for
Re 4000
=
, our results fall within the regions
enveloped by the curves of Borazjani & Sotiropoulos (2008) and Zhou & Shu
(2012) and thus are basically good. From the two force variation curves, we
can see that for both
Re 300=
and 4000, the fish experiences a drag-type
mean net force at low
St
and a thrust-type mean net force at high
St
. The
mean net force variation versus
St
is almost monotonic except in the low
St

region, where the magnitude of the drag-type force generated on the fish first
increases above that of the rigid case, then gradually diminishes, peaking at
around
0.1St ≈
for both
Re

considered. The critical Strouhal
number

critical
St
at which the mean net force transits from the drag-type to
thrust-type decreases with the increase in
Re
.

8.3 Conclusions
The applicability of the new boundary condition-enforced immersed boundary
method to three dimensional incompressible viscous flows with moving
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Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving
Boundary Flows
294

boundaries has been demonstrated in this chapter. Two interesting biofluid
problems involving geometrically complex rigid or flexible bodies are
simulated. The first is a three-dimensional rigid finite-span foil which
undergoes a harmonic heaving and pitching motion in a uniform oncoming
flow. The second concerns a flexible-body fish which commits a
straight-swimming in ocean through an undulatory locomotion along its
flexible body. The aerodynamic or hydrodynamic performances of the foil and
fish and their dependence on various kinematic parameters have been carefully
examined and compared with the established results in the literature, which
show a very satisfactory agreement. The flow fields around these moving
objects, in terms of three-dimensional wake structures, have also been
visualized and matched well with those reported previously. Therefore, the

boundary condition-enforced immersed boundary method offers a powerful
tool for the complex moving boundary problems.

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Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving
Boundary Flows
295



Table 8.1 The variation range of
A
R
and
St


A
R

St

1, 2, 3, 4, 5
0.6
3
0.35, 0.6, 0.8, 1.0

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Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving
Boundary Flows

296










Fig. 8.1 Schematic view of a rigid finite-span foil heaving and pitching
in a free stream






Fig. 8.2 Schematic diagram for the definition of parameters





W
c
U
∞
ψ

h
c
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Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving
Boundary Flows
297


(a)

/0
flap
tT
=


(b)

/1/4
flap
tT
=


(c)
/2/4
flap
tT
=

(d)

/3/4
flap
tT
=

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Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving
Boundary Flows
298


(e)

/4/4
flap
tT
=

Fig. 8.3 Perspective (left column) and side (right column) views of vortex
structures in their
2
λ
-definitions in one flapping cycle


Fig. 8.4 Top view of the vortex structure in its
2

λ
-definition
corresponding to Fig. 8.3(d)


Fig. 8.5 Contours of the spanwise vorticity along the spanwise symmetry
plane for
3AR
=
and
0.6St
=


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Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving
Boundary Flows
299


Fig. 8.6 Contours of the mean streamwise velocity along the spanwise
symmetry plane for
3AR
=
and
0.6St
=

(a)

1AR
=


(b)

2AR
=

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Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving
Boundary Flows
300


(c)
4AR
=



(d)

5AR
=

Fig. 8.7 Perspective (left column) and side (right column) views of vortex
structures for different aspect ratios at
0.6St
=




(a)
1AR =
(b)
2AR
=