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24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Numerical Study on Propulsion by Undulating Motion
in :Laminar-Turbulent Flow
Zuogang Chen and Yasuaki Doi
(Hiroshima University, Japan)
ABSTRACT
Unsteady viscous flow field around a fish-like
advancing body is numerically studied. The main
objective is to study the abilities of a fish-like body to
produce thrust and achieve higher propulsive efficiency
in viscous flow field, and to catch the basic
characteristics of fish-like locomotion. The Reynolds
number based on the oncoming velocity and the body
length ranges from 103 to 3X106. The flow is
simulated by solving three-dimensional unsteady
Reynolds averaged Navier-Stokes equation in a
primitive value formulation, while the eddy-viscosity is
described by algebraic model and transitional zone is
computed by empirical formula. The numerical
scheme is based on the MAC method where the RANS
equation is solved by the time marching method on a
body fitted coordinate system.
The simulations show that pressure is acting as
thrusting force while frictional force is acting as
resistance during the undulating motion. Strouhal
number is the most important governing parameter for
the propulsive efficiency. The propulsive efficiency is
enhanced with the increase of the Reynolds number or
the decrease of body thickness. Two vortices shed
from the tail during one undulating period. Vorticity
travel shows close connection with pressure
distribution. For laminar flow, fish-like locomotion
increases frictional resistance, but for turbulence case,
fish-like locomotion can reduce frictional resistance by
flow relaminarization and achieve a propulsive
efficiency more than 1.
INTRODUCTION
The flow field around a swimming fish or cetacean has
been investigated for a long time by the researchers in
various field of study such as biology, physical science
and engineering. This interest has been inspired not
only to understand and simulate an efficient swimming
propulsion, but also to utilize the results for
engineering application. Fast-starting and
1
maneuvering of flexible hull vehicles can be
significantly better than the performance of rigid
bodies, because the flow can be controlled over the
entire body of the vehicle through the appropriate
flexing. The study of live fish swimming can be very
instructive in exploring mechanisms of unsteady flow
control since fish have evolved over millions of years
to optimize their body shapes and locomotive abilities.
Triantafyllou et al. (2000) pointed out that the
resulting unsteadiness in the flow is exploited by fish
and cetaceans to their advantage. The view is based
on a series of studies demonstrating that the fish (a)
generate large, short-duration forces efficiently, (b)
coordinate rhythmic unsteady body and tail motion to
minimize the energy required for steady propulsion,
and (c) coordinate transient motion of the body and tail
to minimize the energy lost in the wake during
maneuvering. Fish propel themselves through
rhythmic unsteady motions of their body, fins, and tail;
they offer a different paradigm of locomotion that
conventionally used in man-made vehicles. The most
notable proposed mechanisms fall under the categories
of laminar boundary layer maintenance, turbulent drag
reduction, utilization of shed vorticity and the delay of
separation.
As described in Wolfgang et al. (1999), a fish
benefits from smooth near-body flow patterns and the
generation of controlled body-bound vorticity, which is
propagated towards the tail, shed prior to the peduncle
region and then manipulated by the caudal fin to form
large-scale vertical structures with minimum wasted
energy. This manipulation of body-generated
vorticity and its interaction with the vorticity generated
by the oscillating caudal fin are fundamental to the
propulsion and maneuvering capabilities of fish. Muller
et al. (1997) showed that the fish shed one vortex per
half tailbeat when the tail reached its most lateral
position. Part of the circulation shed in the vortices
had been generated previously on the body by the
transverse body wave traveling down the body. The
alternating suction and pressure flows form a
circulating flow around the inflection points of the
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Representative terms from entire chapter:
drag reduction
body; this circulating flow is shed when the inflection
point reaches the tail. Also validated in Ahlborn et al.
(1991), during steady swimming, where vortices may
be generated by the fore-body of a fish, the tail could
either reverse the rotation immediately, or generate an
additional eddy of equal but opposite angular
momentum, so that an eddy pair is produced of zero net
angular momentum. This eddy pair dissipates quickly,
producing a mushroom-shaped flow structure. The
momentum of this flow must be equal to the forward
momentum of the fish. Recovery of most of the
rotational energy and destruction of the eddies allow
the fish to swim more efficiently. In addition, the
destruction of the vortices, or footprints, also makes the
fish less detectable to predators.
Azuma (1992) summed up former researchers'
results and plotted skin friction drag of fish and
cetaceans. In general, the Reynolds number lies in the
range of 103
analyzed tadpole propulsion using a three-dimensional
computational fluid dynamic (CFD) model of
undulatory locomotion that simulates unsteady viscous
flow around an oscillating body of arbitrary
three-dimensional geometry. The simulated results
revealed that the shape and kinematics of tadpoles
collectively produce a small 'dead water' zone between
the head-body and tail during swimming precisely
where tadpole can and do grow hind limbs without
those limbs obstructing flow.
The present paper provides some numerically
simulated results which represent the basic
characteristic of fish-like locomotion. The objective
of present study is to find superiority of propulsion by
undulating motion in laminar and turbulent flow. A
rectangular plate with aspect ratio 1/3 and NACA wing
sections are employed for the simulation. The
phenomena of vorticity control and turbulence
relaminarization are studied. The relationship
between propulsive efficiency and related parameters
such as Strouhal number, Reynolds number and body
thickness are also investigated.
MODELING OF UNDULATION
The simulated body undulates actively in unbounded
oncoming flow. Related variables are normalized by
the body length L and oncoming velocity U. as well as
the time is normalized by L/U. The movement of the
undulating body in y direction is given as
y(`x,t)=axn sint2'zb~x- Sp t)] (1)
where a is amplitude, n=1.1, 2 fib is wave number, Sp
is phase velocity, t is time, where x and y are the
stream-wise and the lateral coordinates whose origin
locates at the leading edge of the body. 0
The transition onsets when the non-dimensional v,
is larger than 14 (Baldwin and Lomax 1978), while the
transitional zone ends where v, is larger than 25.
The governing equations are solved in a primitive
value formulation. A numerical coordinate
transformation is introduced into a body fitted
curvilinear coordinate system to simplify the
computational domain and to facilitate the
implementation of boundary conditions. H-type grid
system is adopted, but near the leading edge the grid is
modified to be adapted for the blunt body head.
The numerical scheme is based on the MAC method
where the momentum equation is solved by the time
marching method on the body fitted coordinate system.
The oncoming velocity and the undulating amplitude
increase smoothly from zero at ~0 to the steady value
at into (to: one specified value). After that the
oncoming velocity and undulating amplitude keep
constant. The first order difference form of the time
derivative is used for an explicit advancement in time.
The convection terms are discretized by the third order
upwind scheme, while all the other spatial derivatives
are discretized by the second order central difference
scheme. On the body surface, no-slip condition is
applied for the velocity. For the pressure, Neumann
type condition is applied to satisfy the momentum
equation. A uniform pressure is applied on the inflow
boundary, while a zero-gradient extrapolation is used
on the outlet boundary. The Poisson equation for
pressure is solved by using Successive Over Relaxation
method.
At Re=103, The computational domain is -3.0—Ax_
5.0, -2.0
The turbulence flow around a flat plate was
numerically simulated at Re=106. Figure 2 shows the
computed y+ vs u+, where u+ and y+ are friction
velocity and wall coordinate respectively. The flow is
laminar at x=0.1091 and transitional at x=0.5933. It
can be found that the velocity profiles show good
agreement with the theoretical solution when the flow
has been fully developed at x=0.8537 and x=0.9494.
30t
.
25
.~20
15
10
5
to
~?
000oooooooooooooooooooooooooo
0 ~ , ~
/ ,:~^ Ad,,-- o x~.1091
D/ ~ ~63' ~ x=0.5933
/~ v x=0.8537
0 x=0.9494
If+
+ - 5.75(logl0Y +5, 5
og,Oy
Figure 2: Computed y+ as u+ for a flat plat at Re=1 o6
The computed drag coefficient for NACA0012 at
Reynolds number 2.8X106 is 6.7X10-3, which shows
good agreement with the experimental result, 6.6X10-3
in Abbott and Doenhoff (1959~. In Figure 3, the
computed pressure coefficients also give agreement
with experimental data (Gregory and O'Reilly 1970~.
-0.5L
nit
A) 0 5
Computed results
Experimental data (Gregory et al. 1970)
02 04 06 08 i
x
Figure 3: Surface pressure distribution for NACA0012 at
Re=2.8x 106
SIMULATION ON AN UNDULATING PLATE
Firstly, the flow around a three-dimensional undulating
plate with aspect ratio of 1/3 is numerically
investigated. Computed results at Re=1000, a=0.1,
b=1.5, n=1.1 are shown in Table 2. When the phase
velocity Sp=1.56, CTX is greater than zero, while
Sp=1.57, CTX is smaller than zero. With the increase
of SO, the frictional force component CFX increases
slightly while the pressure component ~CPX~ increases
sensitively. CTX/CFX is an index to evaluate an error
of the self-propulsion condition. From Table 2, it can
be seen that alp has an error less than 1%. More
accurate alp can be obtained by a linear interpolation
from the computed results. For example, when
Sp=1.5658, alp is interpolated as 0.56843 while the
simulated alp at Sp=1.5658 is 0.56846. In order to
find a suitable Sp and corresponding Hip, this kind of
linear interpolation is reasonably used for the present
study.
Table 2: Sensitivity of alp to CTX/CFX at Re=1000,
a=O. 1, b=1.5, n=1.1
Sp C Fx
1.56 0.1300 -0.1288 0.12x10~2 0.90% 0.576
1.57 0.1 305 -0. 1 3 1 3 -0.8 x 1 0-3 -0.63% 0.563
1.5658 0.1 303 -0. 1 303 0.2X 1 0-4 0.0 1 8% 0.568
Figure 4 shows the simulated history of hydraulic
force coefficients (CFX, CPX, CTX), which are
x-components of force coefficient given by Equation
(6~. In Figure 4, Yend indicates the position of the
trailing edge in y-direction. It is found that the
amplitude of CFX is smaller than that of CPX. The
motion of plate is sinusoidal, however CFX does not
oscillate in the sinusoidal way. The resultant force
CTX oscillates around zero. Because of the symmetry
of the plate movement, there are two crests and troughs
in one undulating period. Figure 5 shows pressure
distributions on the symmetric plane (z=0) when the
thrust reaches the maximum ACES reaches the
maximum at ~2.604) and when the thrust reaches the
minimum ACES reaches the minimum at ~2.707~.
When the thrust reaches the maximum, there are two
zones where the larger pressure difference (red and
blue) acting on both sides of the plate generates thrust.
When the thrust reaches the minimum, there is only
one zone where larger thrust is generated.
Cal
0.15
-A
0.05 A ~ ~ ~ / \ :i
CTO.O5 - V \\1 \J : V V V V
CPX 0O1 ~ ~ 1 ~ ~ ~ ~ /\ ~ ~
-v V V V V V V V
-0.2 1 5 2 ' t 2 5
A ~ ~ ~ ~
.W~.~.~.
1.5 2 t 2.5 3
0.1
0.05
to
-0.05
Figure 4: Simulated time history of hydraulic force
coefficients (the upper) and the corresponding position of the
trailing edge in y-direction (the lower) at Re=1000, a=0.1,
b=1.5, Sp=1.5658
s
1 P: -0.7 -0.6 -o.s -0.4 -0.3 -0.2 -of o ~ ~ ~ 2 ~ 3 1
n 1
O
-0.1
Figure 5: Pressure contours at symmetric plane when thrust
reaches the maximum (the upper) and the minimum (the
lower) atRe=1000, a=0.1, b=1.5, Sp=1.5658
Figure 6 shows time-averaged distributions of
x-components of frictional stress r X' pressure Px and
their resultant stress. The frictional stress acts as a
resisting force on the whole area of plate, while the
pressure acts as a thrusting force. The thrust mainly
comes from the aft part of the plate. Figures 7 and 8
show x-components of frictional force and pressure
contours at t=2.767 (3/81), 2.821 (4/8~, 2.874 (5/81)
and 2.927 (6/81) respectively. It can be seen that
there are two larger pressure zones, one is thrust
(negative zone) and the other is resistance (positive
zone), which move from the middle to the tail
periodically. The distribution of frictional stress does
not vary so much during the undulation.
= === =
-0.15 -0.125 -0.1 -0.075 -0.05 -0.025 0
Figure 6: Distributions of time-averaged x-components of
fiction stress (the upper), pressure (the middle) and resultant
stress (the lower) at Re=1000, a=0.1, b=1.5, Sp=1.5658
0.2 t
- ~ /8~
~ to ^1: ~ ~ ~ ~ C ~ ~ ~ OC A ~ 13 1
0~ _
Figure 7: Distributions of x-components of friction stress at
every 0.125T step when Re=1000, a=0.1, b=1.5, Sp=1.5658
-0.3 -0.2 -0.1 0 0.1 0.2 13/8T|
u o 0.2 0-4 X 0.6 0.8 ~
Figure 8: Distributions of x-components of pressure at every
0.125T step when Re=1000, a=0.1, b=1.5, Sp=1.5658
Dependence of self-propulsion on wave number was
investigated. Table 3 shows the calculated
self-propulsion condition at Re=1000' a=0.1' n=1.1.
In Table 3' A is the maximum lateral excursion of the
trailing edge of the plate and St is Strouhal number,
defined by
St=f xAlU
(9)
where f=b xSp is the frequency of undulation and U is
6
the oncoming velocity. As shown in Table 3, Sp
decreases when b increases to get self-propulsion.
The frictional force depends on the wave number. As
b becomes larger, the frictional force increases. On
the other hand, when b is very small (b=0.5) Sp is so
high that the Fictional force becomes larger. The
time-averaged x-component of friction stress for
various b is compared in Figure 9. It can be seen that
the increase comes from the aft of the plate, where the
larger frictional force zone enlarges with the increase
of b. Figure 10 shows time-averaged x-directional
pressure distribution for various b. It is interesting
that the patterns of the x-directional pressure
distributions are similar, although their wave numbers
are different. The peak of the distribution which
contributes to produce thrust locates at 0.55 Cx<0.85
for each wave number, while the end of the plate does
not contribute to produce thrusting force.
Table 3: Computed results for self-propulsion at
Re=1000, a=0.1
b
0.5
1.0
1.5
2.0
2.5
' Sp
3.10
1.89
- 1.57
1.41
.
1.32
A
0.0988
0.0967
0.0938
0.0906
0.0875
St
0.153
0.183
0.220
0.256
0.289
CFX
0.126
0.125
0.130
0.137
0.143
flat plate
0 0.05 0.1 0.15 0.2 0.25 0.3
LIP
0.369
0.524
0.568
0.558
0.524
l
Figure 9: Time-averaged x-components of friction stress for
various b atRe=1000, GO.1
Figure 10: Time-averaged x-components of pressure for
various b atRe=1000, a=0.1
The calculations for a=0.06, 0.08, 0.12 and 0.14
were carried out. Figure 11 displays the relationship
between a, b and Sp when self-propulsion is achieved.
As the increase of a or b, Sp decreases but it is greater
than 1. Figure 12 shows the dependence of propulsive
efficiency on a and b. It can be seen that there exists
a zone, in which n p can be beyond 0.55, and that A p
decreases gradually when the values of a and b move
away Tom the zone.
.5
06 0.08 0.1
a
0.12 0.14
Figure 11: Dependence of Sp on a and b
7
225t
2
1.75
1.5
1.25
1
0.75
~ _
0.06 0.08 0.1 0.12 0.14
a
Figure 12: Contours of propulsive efficiency
From Figures 11 and 12, it can be observed that alp
seems to depend on a, b and Sp. These three
parameters are combined in one parameter St (Strouhal
number) as shown in Equation (9~. Figure 13 shows
the dependence of the propulsive efficiency alp on
Strouhal number St. The curves of alp concentrate
within a narrow range so that, for an undulating plate,
the governing parameter for the propulsive efficiency is
not the amplitude, wave number or phase velocity, but
Strouhal number. The highest alp is about 0.58 when
St is around 0.23. The higher alp, which is greater
than 0.55 for example, is achieved at 0.2< St< - 0.26
when Reynolds number is 1000.
0.6 _
0.55
so 0.5
.O
~ 0.45
c'
<,, 0.4
0.35
0.3
. a=0.06
· a=0.08
a=0.1 0
· a=0.1 2
1 · a=0.14
0.15 0.2 0.25 0.3 0.35
Strouhal number
Figure 13: Dependence of propulsive efficiency on Strouhal
number
SIMULATION ON A TWO-DIMENSIONAL
UNDULATING NACA WING
Because Strouhal number was verified as the important
parameter to achieve high propulsive efficiency at one
chosen Reynolds number, the emphasis of this section
is given on the dependence of propulsive efficiency on
the body thickness and Reynolds number, as well as
basic characteristics in fish-like locomotion. Like the
simulation in the last section, the parameters are still
chosen as a=0. 1, b=1.5, n=1. 1 for Re=1000.
The basis for enhancing performance through
unsteady flow control is the formation of large-scale
vortices through body motion, the sensing and
manipulation of these vortices as they move down the
body, and the eventual repositioning through tail
motion. These concepts constitute the essence of
vorticity control. Figure 14 displays pressure
distributions and streamlines around NACA0010 at
Re=1000, a=0. 1, b=1.5, Sp=1.78. The obvious
vortices have been generated at concave parts with the
undulation of the body and travel down the body before
they shed into the wake. The propagating undulation
improves the pressure exerted on both body sides.
The high pressure zone (indicated by red) travels
simultaneously with the vortices down the body.
Meanwhile from the figure, it is obvious that high
pressure zone and low pressure zone always exert on
both sides of the body and act as thrust, while in the
wake the pressure is lower in every center of vortex.
It shows the body sheds one vortex per half period
when the tail is near its most lateral position (0/81~.
The vortices shed from the tail and fade in the wake.
They look like reverse Karman street. Thus the
presence of vorticity in the wake of an undulating body
in viscous fluid is a consequence of the need for a
propulsive jet to counter the body drag.
The relationship between the propulsive efficiency
and body thickness was investigated. Figure 15
shows the computed Sp and alp for NACA wing with
different relative thickness. The increase of body
thickness leads to the increase of phase velocity and the
decrease of propulsive efficiency, which implies some
advantages for slender swimmers.
Laminar and turbulent flow around an undulating
NACA0010 was investigated. Figure 16 shows the
variations of propulsive efficiency and frictional force
coefficient. CFO is the frictional force coefficient for
rigid body while CFX is that for undulating body at
self-propulsion state. At low Reynolds number, CFX
is over CFO by more than 70%, but at enough higher
Reynolds number, CFX is smaller than CFO Table 4
lists the values of drag reduction and propulsive
efficiency for NACA00 10 at a=0.05 1, b=1.0.
Frictional resistance reduction is about 22% at
Re=2xlo6 and about 32% at Re=3xlo6. It is found
that the propulsive efficiency defined by Equation (8)
is greater than 1 at Re=2xlo6 and Re=3xlo6. It
numerically shows that the power required to propel an
actively swimming fish-like body is smaller than the
power needed to tow the body straight and rigidly at
the same speed.
8
'.9L
~.8 -0.7 ~.6 ~.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
no
in,
-0.5
n ~
n is
1.8
1.7
1.6'
0.6
0.58
0.56 ~
c,
0 54 Ha
: 0.52
6 ~ 10 12 14 1 60.5
Relative percentage thickness
Propulsive efficiency /
~ /
--^ ~
~ - -^
— -A
Figure 15: Dependence of Sp and Alp on body thickness at
Re=1000, a=0.1, b=l.S
O O CFO for rigid body
O O CFX for unduladog body
_ G · Propulsive efficiency
~ ° 0 ~ .
.~1.2
Figure 14: Pressure distributions and streamlines for
NACAOO10 atRe=103, a=0.1, b=l.S, Sp=1.78
9
0.11
0.06
x
,~
A
Lo
0.01
.
.
' . . .
103 104 105 1o6
Reynolds number
1 . 1 c,
1 a)
.O
0.9=
a)
0.8 .>
_
0.7 Q
0
0.6 Q
- c
_ 0.5
Figure 16: Variations of fictional force coefficient and
propulsive efficiency with Reynolds number for NACAOO 10
Table 4: Drag reduction and propulsive efficiency for
NACAOO10 at a=0.05 1, b=1.0
Figure 17 shows the distributions of unsteady local
friction stress coefficients on the upside body surface
during one undulating period at Re=2x 106. The
thicker line plots the friction stress for a rigid body.
In the figure, x=0 and x=1 represent the leading and
trailing edge respectively. As shown in Figure 17, the
friction stress on undulating body surface obviously
diminishes on most part of the body (0.3_x_0.9~.
Figure 18 plots the calculated transition zone. During
a period, the laminar flow domain is enlarged by the
undulating motion, which implies that the fish-like
undulation delays the turbulence transition. Figure 19
compares the velocity profiles at x=0.1416, 0.3510 and
0.7617. The thicker lines plot the velocity profiles for
a rigid body, while the marks represent the unsteady
profiles during one undulating period. The dashed
line plots the velocity of logarithm law. It is found, at
x=0.1416, the undulation makes the profiles deviate
laminar form in a small degree, which increases local
frictional stress as shown in Figure 17. While at
x=0.35 10, the undulation makes the turbulent flow
relaminarize at some instants. At 318T, 4/8 T and 518T,
the profiles deviate from logarithm law form, so the
Fictional force is reduced greatly as shown in Figure
17. Meanwhile, the profiles at x=0.7617 are close to
the logarithm law, but the undulation decreases the
magnitude of the friction stress.
O.On3
c
o
0.2 0.4 0.6 0.8 1
X
4n
30
20
10
o
rigid
0 0/8T
1/8T
2/8T
D 3/8T
~ 4i8T ~20
a A/RT
30
10
an
Figure 17: Distributions of local friction stress coefficient for
NACAOO10 at R~2X 1 o6
1n
o.s
0.8
0.7
0.6
~0.5
<0.4
0.3
1
0.2
0.1
O—
0/8
The area between two dashed lines:
transitional zone of rigid body
The area between two solid curves:
t~anstional zone of undulating body
,
1/8 2/8 3/8 4/8 5/8 6/8 7/8 0.5
T
Figure 18: Transition zone variations caused by undulation at
Re=2x 1 o6 for NACAOO10
Figure 20 plots the variations of pressure gradient in
x-direction at the mesh points most near the upside
body surface. The undulation reduces the mean
pressure gradient at almost whole zone, which may be
the main reason for turbulence suppression. The
corresponding flow phenomena can be observed in
Figures 21 and 22, which respectively plot the
distributions of vorticity ~ co ~ and relative velocity uref at
the mesh points most near the upside body surface.
The relative velocity uref denotes x-component velocity
subtracted by velocity component on the body surface.
From Figure 21, globally the vorticity becomes smaller
under undulation, which implies that turbulence
strength is also reduced because the eddy viscosity is
bound to diminish by mixing length formula (Baldwin
and Lomax 1978~. From Figure 22, when x ranges
between 0.3 and 0.9, the relative velocities Uref are
smaller than those for rigid case. The decrease of
relative velocity leads to the reduction of friction stress.
inner ,0~ o o ~ a o o
0/8T X=0.4446 / t/\ ~ i'
2/3T / ~
4/8T /°~ ----a
5/8T it -----------
7/8T_ - f
v
D
O
O
*
1 IO910{ 2
G
040
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
ngid
· 0/8T
t · 1/8T
2/8T
- ~ 3/8T
4/8T
it. r 0 5/8T
'id
0 0.2 0~4 X 0.6 0.8 1
Figure 22: Comparisons of relative velocity at mesh points
most near the upside body surface
The computations for different phase velocity were
performed. Figure 23 shows the variations of
hydraulic coefficients with phase velocity for
NACA0010 at Re=2xl06, a=0.051, b=1.0. When
Sp=1.2, c TX becomes zero so self-propulsion is
achieved. When C TX iS larger than 1.2, the net
thrust can be obtained. The present study evaluates
the propeller efficiency when the flexible-hull body is
taken as a propulsor. The related propeller efficiency
~7 is defined as
J outputdt / T U |TXdI
q= r = T (10)
|inpu~dt/T J.[| (a +Px) ups+ ~ (ry +py) vds]~
T IS IS
where the input work rate is defined as same as that in
Equation (8) for up, the output work rate is the
product of net thrusting force Tx and oncoming velocity
U. Certainly ~ is zero for self-propulsion state, just
like case 1 in the Table 5. Cases 2, 3 and 4 represent
the computed results for NACA0010 at different phase
velocity. With the increase of SP, ~ and the thrust
coefficient CTX increase monotonously when SP
ranges from 1.2 to 1.6. The difference between case 4
and case 5 is the body thickness. The thinner
flexible-hull can achieve higher efficiency and generate
larger thrust. The comparison between cases 5 and 6
explores that ~7 increases with Reynolds number.
The high efficiency of ~ shown in Table 5 indicates
that fish-like undulating body can be used as a
propulsor because of its satisfied propulsive
performance at higher Reynolds number.
0.0z ,
1~ 0.01
~ .
1C)
1C) -0 01
~ Ado_ A_
_ ______ _1____~ _ ~~ __ _ L _ _
I \ \ I
_ _ _ _ _ _ _ A_ _ _ _ _ _ _ _ _ _~ _ _~ _
~ 1 -9
_ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ _ _ _ _ _ _ >~< _
I 1~
. . , . . . . . . . . . . , · ~
0 05 Sp 1 1.5
Figure 23: Hydraulic coefficients with phase velocity for
NACAOO10 atR~2xlo6'a=0.051, b=1.0
Table 5: Investigation on propeller efficiency
Case
_
2
-3
4
5
6
Re
2xlo6
2xlo6
2xlo6
2xlo6
2X106
3X106
.-
Relative
thickness
10%
10%
10%
10%
_ O
Sp
.2
.3
1.5
1.6
1.6
_ 1.6
CTX x 1 0
.
-0.001 1
-0.321 1
-0.9986
-1.353
-1.935
-1.981
o
0.2992
0.5019
0.5389
0.6264
0.6438
SIMULATION ON THE THREE-DIMENSIONAL
UNDULATING BODY
The two-dimensional results were expended to
three-dimensional cases by setting the body width is
0.24. At the plane of z=0, the body section is NACA
wing, while the body thickness is zero on the fringes
(z=+0.12~. The body thickness varies smoothly at
every x=const. plane. The body is called as "flat fish"
in the present study.
The simulated results at Re=103 are listed in Table 6.
When the computed result of 3D (case 8) is compared
with 2D (case 7), both the phase velocity and frictional
drag increase becomes smaller. That the propulsive
efficiency becomes small shows that the 3D effects
weaken the propulsion performance. The prediction
is same to the conclusion by Liu et al. (1997), whose
simulated object was a swimming tadpole. Figure 24
shows three transverse cuts (parallel to yz axes),
demonstrating that the flow varies along the
longitudinal direction. The longitudinal vortices exist
both near the body and in the wake. Probably the
strong rotating flow has significant effects on
propulsive efficiency for 3D case. Also, like the
conclusion drawn in two-dimensional case, the increase
of body thickness leads to the increase of phase
velocity and the decrease of propulsive efficiency (see
cases 8 and 9~.
Table 6: Investigations on self-propulsion for the flat
fish at Re=103, a=0. 1, b=1.5
.
Case
7
8
9
Maximum
relative thickness
10% (2D)
10% (3D)
14% (3D)
T:
1.761
1.610
[ 1.694,
I ( C Fr / COO) - 1
1 75.7%
l
1 65.0%
T 76.4%
1 UP 1
1 1
1 0.547
T 0.4581
1 0.408 1
The simulated results for a flat fish with maximum
body thickness of 14% at Re=2x105 are listed in Table
7. Turbulent flow simulation shows smaller frictional
drag increase and higher propulsive efficiency, which
implies again that the undulation can suppress
turbulence. From cases 10 and 9, the propulsive
efficiency enhances with the increase of Reynolds
11
number. One phenomenon is shown in Figure 25, in
which x-component of pressure stress is compared for
the two cases. Pressure acts as thrusting force, but for
lower Reynolds, the time-averaged pressure on a small
area near the trailing edge is not thrust but resistance.
Table 7: Comparison on flow state for a flat fish with
maximum body thickness of 14% at Re=2x 105,
a=0.067, b=1.087
Case Flow state SP (Cry/ CFO)-1 ~ P
10 Laminar 1.278 55.9% 0.568
1 1 Turbulent 1.287 17.5% 0.662
~ i , .~! :~.: ~~ ~~.~; :: ~ :~:
~1 Pressure: ~.7 -0.6 -0.5 -0.4 ~.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 `~ it- ~~.~
Figure 24: Pressure and velocity distributions along planes
perpendicular to the advancing direction of a flat fish with
maximum thickness of 10% at Re=103, a=O. 1, b=1.5, Sp=1.61
n 1
0.2
n 1
n-
O ~ ~ . . i . , . . _
0 0.2 0.4 0.6 0.8 1
X
~ () 0.2 0.4 X 0.6 0.8 1
Figure 25: Time-averaged x-component of pressure
distributions atRe=103 and 2XlOs for a flat fish
CONCLUSIONS
The MAC method with a body fitted coordinate
system is applied to simulate the laminar-turbulent
flow around an undulating advancing body. A
modified turbulence model is adopted for Reynolds
averaged Navier-Stokes equation. The simulation
shows that pressure is acting as thrusting force while
frictional force is acting as resistance during the
undulating motion. Thrust goes up with the increase
of undulating wave propagation velocity. Strouhal
number is the most important governing parameter for
the propulsive efficiency. The propulsive efficiency is
enhanced with the increase of Reynolds number or the
decrease of body thickness. For laminar flow,
frictional force enlarges with the increase of phase
velocity. But for turbulent flow, fish-like locomotion
can reduce frictional resistance by turbulence
relaminarization and achieve a propulsive efficiency
more than 1. Two vortices shed from the tail during
one undulating period. Vorticity travel shows close
relation to pressure distribution. Fish-like undulating
body shows the potential as a propulsor.
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