flaps shown are higher. There is a

flaps shown are higher. There is a general agreement with Triantafyllou’s
@13# experiment on a two-dimensional heaving and
feathering foil. It is interesting to note that the dual flaps show a
tendency to achieve a higher efficiency in the waving mode. Because
the nose exhibits a yawing oscillation in the waving mode,
it is hypothesized that this sheds vortices which lowers the drag
on the rigid body due to boundary layer interaction, or it enhances
the thrust ~by augmenting jet speed! due to the vortices produced
by flapping. This line of thinking led to the second experiment.
The efficiency plots in Fig. 18 include the cylinder drag. The
viscous and form drag coefficient of the cylinder is 0.145. When
this is taken into account, the efficiency of the flapping foils alone
are higher as shown in Fig. 19. At lower values of St, efficiency
has a stronger dependence on f.
All measurements of axial force coefficient due to single and
dual-flapping tails are shown in Fig. 20. Both thrust and drag
producing cases are included. The trend displays a sensitivity to f.
At f52.6 and 6.2 Hz, the single foil does not produce a net thrust.
However, it does at f54.24 Hz, where the single and dual flaps in
both modes follow a similar trend. Tripping of the cylinder
boundary layer has no effect on the thrust produced. The data
indicate that thrust produced is governed by St, f, and number of
flaps. Mode of flapping has a minor effect. According to Triantafyllou
~Pvt. Comm. 1997!, the ‘‘robotuna’’ vortex cores make
an angle of 10–15 deg to the forward direction. However, the
wake angle is 140 deg in the present case. The wider wake growth
in the present case of a rigid cylinder requires a closer examination.
3.3.3 Sensitivity to Strouhal Number and Flapping Frequency:
Single Foil Case. Figure 21 shows the time-averaged
coefficients of axial force and pitching moment for one single foil
attached to the rigid body in the presence of the dividing plate.
The coefficients do not depend solely on St, they also depend on
f. The sign of the axial force and yawing moment change when
f54.24 Hz. For the single flap, the higher sensitivity of ca and cm
to St at f54.24 Hz can be further demonstrated by examining the
unsteady behavior. Figures 13 and 14 show that a peculiar aspect
of the fish propulsion and maneuvering mechanism is the fact that
large unsteady forces are produced to generate a range of timeaveraged
levels. We believe that this unsteady behavior holds the
key to its propulsion and maneuvering mechanism and the longtime-
averaged values do not clarify this. The maximum and minimum
values within a cycle of the time signatures of ca and cm(y)
are expressed as
y5ea~St!n, (11)
where the exponents a and n are characteristics of f. The sensitivity
Q is then given as
Q5nea~St!n21. (12)
The values of Qa and Qm are calculated at St50.3 as shown in
Fig. 22. The sensitivity of the unsteady mechanism is highest at
f54.237 Hz. The cause of this sensitivity to f is not well understood.
Triantafyllou and coworkers have not examined moments
and have not noticed such dependence on (St, f ). We propose that
an interaction between the vortex shedding from the cylinder and
the flapping foil is causing an instability in the vortex train. Another
Strouhal number involving the length of the cylinder, amplitude
of its head swaying, and f are likely involved. The result is a catastrophic switch from a regular Karman train to a negative
Karman train. Further work is needed to verify this hypothesis.
3.4 Vortex Shedding: Vorticity-Velocity Vector Maps
3.4.1 Vorticity-Velocity Vector Maps. The vorticity-velocity
vector measurements of the vortex shedding process from the tail
flapping foils, phase matched to its motion, were carried out at a
flow speed of 20 cm/s. Their maps in the axial ~diametral! midplane
(z50) are shown in Figs. 23, 24, and 25 for clapping,
waving and clapping modes, respectively ~phase is given by t*
5tU` /D)d50. Similarly, the phase-matched vorticity-velocity
vector maps in the cross-stream plane at the trailing edge of the
flap (x/D50.066) are shown in Figs. 26 and 27 for the waving
and clapping modes, respectively. Such maps were used to compute
circulation values of the vortices by two methods: by calculating
velocity line integrals and vorticity area integrals. The distributions
of circulation of the axial vortex generated at the flap tip
are shown in Figs. 28 and 29 for x/D50.0656 and 0.5577, respectively.
The two methods of circulation calculation, based on
vorticity-area and velocity-line integrals, are in reasonable agreement.
Note that within a short length after formation (x/D
>0.5), the absolute value of the minimum circulation has
dropped by a factor of 3. Measurement resolution is higher in Fig.
23. This figure captures the radially far-flung vortices. The maps
in Figs. 23, 24, and 25 show the jets between vortex pairs which
give rise to thrust. The information in Figs. 26 and 27 has been
used in Fig. 30 to depict the trajectory of the axial vortex schematically
and the effect of the divider on it. The vortex arrays and
the mechanism of thrust and yawing moment are depicted schematically
in Figs. 33 and 34, for clapping and waving modes,
respectively.
Figures 24 and 25 indicate that, in the clapping mode, the two
flaps produce arrays of vortices that are mirror images. They produce
a net thrust but no net maneuvering cross-stream forces ~Fig.
33!. On the other hand, in the waving mode, the two arrays of
vortices from the two flaps are staggered in the streamwise direction.
Due to this fact, the waving mode produces both axial and
cross-stream forces ~Fig. 34!. The vortex shedding process is