Strouhal numbers and optimization of swimming by odontocete cetaceans
1 SSC San Diego, 53560 Hull Street, 211, San Diego, CA 92152, USA
2 Department of Biology, West Chester University, West Chester, PA 19383,
USA
* Author for correspondence (e-mail: rohr{at}spawar.navy.mil)
Accepted 17 February 2004
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Summary |
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Key words: Strouhal number, swimming, dolphin, cetacean, odontocete, Tursiops, Pseudorca, Orcinus, Lagenorhynchus, Globicephala, Stenella, Delphinapterus
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Introduction |
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A similar conclusion, but through more rigorous theoretical analysis and
detailed experimental studies, has been drawn by Triantafyllou et al.
(1991,
1993
). Performing stability
analysis of the mean velocity profiles of a pitching airfoil, Triantafyllou et
al. (1991
,
1993
) have shown that maximum
spatial amplification and optimum creation of thrust-producing jet vortices
lies in a narrow range of nondimensional frequencies referred to as the
Strouhal number (St). The predicted St range for maximum
spatial amplification occurs between 0.25 and 0.35, peaking at 0.30
(Triantafyllou et al., 1991
,
1993
;
Streitlien and Triantafyllou,
1998
). Triantafyllou and Triantafyllou
(1995
) have argued that for
St=0.250.35, swimming efficiency for fish and cetaceans would
also peak. Experiments with isolated oscillating foils have found highest
propulsive efficiencies for St between 0.20 and 0.40 (Triantafyllou
et al., 1991
,
1993
,
2000
;
Anderson et al., 1998
;
Read et al., 2003
).
The St, which is related to how fast the vortices are being
generated and the space between them, is defined as:
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Although many species of cetaceans are believed to be particularly swift,
efficient swimmers, corresponding St data have been extremely
limited. The cetacean Strouhal number data of Triantafyllou et al.
(1993) consist of only two
values. Moreover, they were indirectly derived from analysis of traces
(Lang and Daybell, 1963
)
obtained from motion picture frames of a 2.03-m Pacific white-sided dolphin
(Lagenorhynchus obliquidens) swimming non-uniformly within a tank.
One St value, 0.32, corresponded to the dolphin swimming while
wearing a 1.91 cm-diameter drag collar. The remaining St of 0.30
corresponded to swimming without the drag collar. References to this less than
optimal data set (Triantafyllou et al.,
1991
,
1993
,
2000
;
Triantafyllou and Triantafyllou,
1995
; Taylor et al.,
2003
) appear repeatedly throughout the literature.
Many different species of captive odontocete cetaceans have been trained to swim steadily behind viewing panels and provide a unique opportunity for a much larger, more accurate St database. Here, Strouhal numbers are calculated from recordings of six species of trained odontocete cetaceans. The species include members of the family Delphinidae: bottlenose dolphin (Tursiops truncatus), false killer whale (Pseudorca crassidens), spotted dolphin (Stenella frontalis), striped dolphin (Lagenorhynchus obliquidens), killer whale (Orcinus orca) and pilot whale (Globicephala melaena). For comparison, Strouhal numbers for the slower, less efficient swimming beluga whale (Delphinapterus leucas; family Monodontidae) are also included but treated separately.
The objective of the present study was to investigate the range of Strouhal
numbers preferred by swimming cetaceans, how this range varied between
species, within species and for individuals and, most critical, what range of
Strouhal numbers corresponded to maximum propulsive efficiency. Data directly
relating swimming efficiency to St, for any swimming animal, have
previously not existed (Bandyopadhyay et
al., 2000).
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Materials and methods |
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Experiments were performed in large elliptical pools with maximum lengths of 27.448.8 m. The curved portions of each pool were constructed of 1.72.1 m-wide Plexiglas panels separated by 0.2 m-wide posts, allowing for an unobstructed view of the animals as they swam. A water depth of 1.42.1 m was visible through the panels. The depth of the pools was 7.311.0 m, and water temperature ranged between 12°C and 22°C.
The animals normally swam about 0.51.0 m below the surface of the water and 1 m from the pool walls. Many of the animals were trained to perform `fast swims' around the circumference of the main performance pool, thereby providing a larger range of swim speeds to study. Initial training involved instructing the animals to accelerate quickly and touch their rostrum to a boat pole held several meters in front of them. Through successive approximations of positioning the boat pole further and further away, the dolphins were trained to swim fast upon command. When the dolphins were not swimming at what the trainers judged to be peak performance, the boat pole was slapped on the surface of the water to induce the dolphins to swim more rapidly. Trainers frequently raced animals in pairs to provide additional incentive. A wide variety of rewards, including tactile stimulation, environmental enrichment devices and food, were given for appropriate behaviors on an intermittent reinforcement schedule. In addition to performing in shows, the dolphins in this study participated in training, play, relationship, husbandry and exercise sessions on a regular basis. Approximately 1820 h of their day consisted of nonstructural play, free and rest time.
Video analysis
A camcorder (Sony CCD-TR81 or Panasonic DV-510) was used to record swimming
sequences of cetaceans at a rate of 30 frames s1. The
camcorder was positioned in front of the Plexiglas wall of the pool, allowing
for a clear view of three panels. Swimming motions of the animals were
recorded as they routinely swam and when they were encouraged to swim at
maximum speed. Sequential body and fluke positions were determined directly
from individual frames of videotape with a Panasonic AG-7300 video recorder
and video monitor or were digitized using the Peak Motus video analysis system
(version 4.3.1; Peak Performance Technologies, Englewood, CO, USA). Kinematic
data from video records to calculate Strouhal number included mean swimming
speed (U; m s1), fluke oscillation frequency
(f; Hz) and peak-to-peak fluke amplitude (A; m). A
is defined as the maximum vertical displacement of the trailing edge of the
flukes. Only video sequences in which the animals appeared to be swimming
horizontally and at a constant speed were used.
At Sea Worlds in Orlando and San Antonio, the animals were marked with zinc oxide reference points on the lateral aspects of the caudal peduncle. Marks were separated by a measured distance of 0.10.39 m and served as the scale for video analysis. At the San Diego site, the animals were not marked. The scale was determined from a marked section of the Plexiglas panels of the pool. U was determined by dividing the length of a marked section through which the dolphin swam by the time that it took the dolphin to swim across it. Time was determined from the frame rate. Dolphin swimming speed measurements obtained from the video could be accurately repeated to within a few percent. This uncertainty resulted from the fact that time was quantified by the frame rate, so crossing points could be off by a fraction of a frame rate. At the highest speeds of 8 m s1, the 8 m run resulted in 30 frames between start and finish, of which the last two frames were suspect. This could result in, at most, a 7% uncertainty (two frames out of 30).
To assess if the Plexiglas panels and the recording position affected the
calculations of U, video recordings of a cast model of a dolphin
dorsal fin were made as it was moved along the normal swimming trajectory of
the animals. The difference in distance between the actual positions where the
cast fin crossed the reference marks and that determined from the video
recordings was insignificant. f was calculated by dividing the frame
rate by the number of frames comprising a single complete oscillation of the
tail. Again, the frame rate limited accuracy for determining the tail
oscillation period. For a relatively high frequency of 3 Hz, the period was
determined from 10 frames. With a full frame ambiguity at each end of the
oscillation the uncertainty would be, at most, 20%. The peak-to-peak amplitude
of the fluke, A, along with a previously measured reference length
marked on the tank wall, was measured directly on the television monitor
screen. The reference length was recorded inside and outside the pool to
account for refraction effects. The reference length provided a means of
converting lengths measured on the monitor screen to actual distances in
meters. The spatial resolution of the fluke at maximum and minimum amplitude
is, as for the swim speed measurements, affected by the framing rate of the
camera. However, because the fluke tip is moving relatively slowly at these
extremes, the majority of the amplitude uncertainty resulted from resolving on
the video screen the position of the tip of the fluke, particularly if the
fluke was close to the water's surface. When Strouhal number calculations were
repeated for the same recordings but by independent observers, swimming speed
and tail beat frequencies showed excellent agreement; for tail beat
amplitudes, differences of 1020% were not uncommon. This uncertainty
resulted from insufficient screen resolution, framing rate and the proximity
of the tail to the water surface. Overall St uncertainty was
estimated to be 20%. This is 6% less than the St uncertainty
calculated for the worse case scenario by propagating the independent
uncertainties estimated for high speeds (7%), high frequencies (20%) and a
fluke amplitude uncertainty of 15%.
To adjust for size differences between species, data were analyzed with
respect to length-specific velocity (U/L) and
length-specific amplitude (A/L). In some analyses,
f was non-dimensionalized by dividing frequency by
U/L. Means were calculated for values that did not vary with
L or U/L. Variation about means was expressed as
±1 S.D. Linear relationships were estimated by
least-squares regression (Microsoft Excel). Differences in slopes of the
regressions were analyzed by analysis of covariance
(Zar, 1984). Means of species
were compared using analysis of variance (ANOVA; Statistica Version 4.1,
StatSoft). Results were considered significant at the
=0.05 level.
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Results |
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With no statistically significant correlation, A/L was
found to be relatively insensitive to both U/L
(Fig. 1A) and f
(Fig. 1B) for all species. The
mean value of A/L for all odontocetes was 0.21±0.03
(n=267). Mean A/L ranged from 0.25±0.02 for
S. frontalis to 0.17±0.02 for G. melaena with 89% of
the data residing between 0.15 and 0.25. ANOVA showed that there was a
significant difference for A/L among species
(P<0.001; F=9.76; d.f.=6, 260). Aggregating all the
odontocete data (n=267), f was found to increase linearly
with increasing U/L as
f=0.89(U/L)+0.59 (r2=0.8;
Fig. 2). A positive linear
relationship between f and U/L is similar to
results reported for cetaceans, fish and other marine mammals
(Bainbridge, 1958;
Hunter and Zweifel, 1971
;
Webb and Kostecki, 1984
;
Feldkamp, 1987
;
Fish et al., 1988
;
Scharold et al., 1989
).
Regression equation for f and A/L with respect to
U/L for each species is provided in
Table 2. The negative slope in
the regression equation for f by G. melaena is due to the
limited speed range.
|
|
|
Strouhal data
The computed Strouhal number showed little dependence on body length or
swim speed for the delphinid species (Fig.
3). Aggregating animals for each species
(Fig. 4), mean St
values generally reside near the lower boundary of the 0.250.35 range
(Table 3) predicted by
Triantafyllou et al. (1991,
1993
) for peak propulsive
efficiency. Excluding D. leucas, the mean St for the
delphinids was 0.26±0.05 (n=248). The predicted
0.250.35 St range captured 55% of the delphinid St
data (Fig. 5), whereas the
range from 0.2 to 0.3 contained 74%. For an incremental St range of
0.05, the majority of the data were found between 0.225 and 0.275 (44%).
D. leucas had a mean St of 0.35±0.10 (n=19),
which was conspicuously higher than most of the St values for the
delphinids.
|
|
|
|
Propulsive efficiencies, which were previously reported by Fish
(1998), are plotted as a
function of St in Fig.
6. For P. crassidens, O. orca and T. truncatus,
propulsive efficiencies were found to broadly peak at about 0.90, 0.87 and
0.85, respectively, over a relatively narrow range of St
(0.230.28). Outside this St range, where measurements exist,
efficiencies drop off rapidly. The St range favored by P.
crassidens, O. orca and T. truncatus was within this same range,
0.225<St<0.30 (Fig.
7AC). The efficiency data for D. leucas were lower
(0.83) and exhibit a conspicuously broader peak at
St=0.250.40. The distribution of D. leucas St was
relatively flat, with a narrow peak occurring at St=0.4250.45
(Fig. 7D).
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Discussion |
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The data for all delphinids (n=248) showed little dependence on
St over a range of swim speeds from about 28 m
s1 (Fig. 3B).
It has been hypothesized for cruising flight and swimming that St
would be `tuned' for high propulsive efficiency (Triantafyllou et al.,
1991,
1993
;
Taylor et al., 2003
). Cruising
speeds for the cetaceans have been reported from
15 m
s1 (see Fish,
1998
). For the present St data
(Fig. 5), a conspicuous peak
was not apparent at cruising speeds (Fig.
3B) or where maximum propulsive efficiency was predicted
(St=0.250.35). Moreover, the St data were not most
concentrated in the predicted range. Whereas 55% of the data fell within the
predicted range of St=0.250.35, 74% of all the St
data occurred between 0.2 and 0.3. Some of the scatter in the St data
is a result of measurement uncertainty. Wolfgang et al.
(1999
) have reported
St uncertainties of
30% for studies with fish (Danio
malabaricus). However, a large part is presumably due to natural
variation of the animal's swimming motion
(Rosen, 1959
;
Wolfgang et al., 1999
). Kayan
and Pyatetskiy (1977
) reported
a dependence of St on acceleration for captive T. truncatus,
with St increasing with increasing acceleration. Taylor et al.
(2003
) similarly found that,
for birds, St was significantly higher for intermittent as opposed to
direct flight. Although data from the present video analysis were limited to
steady swimming speeds, effects due to small accelerations were possible. A
dependence of St on acceleration may partly explain the difference in
St values for L. obliquidens between the present data
(St=0.24±0.03, n=17) and those inferred from traces
of an accelerating animal (St=0.30, n=1; Triantafyllou et
al., 1991
,
1993
).
One cannot be certain that the mode of steady swimming in captivity for
relatively short durations and near the water surface is similar to that
employed for long durations in the wild. However, the high-speed swimming
capability of regularly exercised captive and free-ranging dolphins is
generally similar (Rohr et al.,
2002). Pershin
(1988
) reports an St
value of 0.37 for a free-ranging dolphin, which he refers to as a common
dolphin. Pershin (1988
) makes
no reference to whether the animal was accelerating or how the recordings were
made. Unlike the captive dolphins in the present study, this free-ranging
dolphin was not swimming near the surface. Except for D. leucas and
S. frontalis, an St value of 0.37 is conspicuously higher
than the mean values reported here. It is not known if this disparity reflects
differences between species, captive and free-ranging animals, steady or
accelerated swimming or different depths beneath the surface.
A hydromechanical model of lunate-tail propulsion based on
three-dimensional unsteady wing theory with continuous loading
(Chopra and Kambe, 1977;
Yates, 1983
) was used by Fish
(1998
) to calculate efficiency
for O. orca, T. truncatus, P. crassidens and D. leucas.
Here, efficiency refers to the Froude efficiency defined as the mean rate of
mechanical work derived from mean thrust, divided by all the work that the
animal is performing while swimming (Chopra
and Kambe, 1977
). The calculated efficiencies are similar to
values reported previously using different hydrodynamic models
(Fish, 1998
). Maximum
efficiencies of 8590% generally occurred at typical cruising speeds
(Fish, 1998
). Similarly, the
minimum cost of transport coincides with the range of cruising speeds in
T. truncatus (Williams et al.,
1992
; Yazdi et al.,
1999
). Efficiency values as high as 85% have been measured for
advanced torpedo propellers that take advantage of the boundary layer wake
(Lang and Daybell, 1963
).
Dolphin efficiencies were significantly higher than values measured for
conventional small rigid propellers, which are typically no more that 40%
(Triantafyllou and Triantafyllou,
1995). The higher efficiencies of dolphins are probably due to
some degree of St `tuning' (in 0.2 St increments, 94% of the
St data were between 0.2 and 0.4), oscillatory fluke motions and
greater flexibility of the flukes. Oscillating foils can produce high lift
coefficients and efficiencies by vorticity control
(Gopalkrishnan et al., 1994
;
Anderson et al., 1998
;
Ramamurti et al., 2002
;
Read et al., 2003
). The
unsteady effects can increase the lift and permit the foil to function at high
angles of attack without stalling. Cetacean flukes are lateral extensions of
the tail and are a composite of flexible elements, including dense fibrous
tissue, ligaments and blubber (Felts,
1966
). The only rigid components in the cetacean tail are a series
of short caudal vertebrae, which support the flukes axially
(Rommel, 1990
). The
white-sided dolphin (Lagenorhynchus acutus) shows 35% and 13%
chordwise and spanwise deflections, respectively
(Curren et al., 1994
).
Chordwise flexibility of an oscillating foil has been demonstrated
experimentally and theoretically to increase efficiency by up to 36% with only
a small reduction in thrust compared with a rigid foil (Katz and Weihs,
1978
,
1979
;
Bose, 1995
;
Prempraneerach et al., 2003
).
Heaving and pitching motions of flexible foil produced a maximum efficiency of
0.87 at St=0.3, and an efficiency of 0.8 was achieved at various
combinations of angles of attack around 15° and St ranging from
0.17 to 0.35 (Prempraneerach et al.,
2003
).
Prior to the present investigation there have been no studies, for either
cetaceans or fish, that addressed whether swimming efficiency occurs within
the predicted 0.250.35 range of Triantafyllou et al.
(1991,
1993
)
(Bandyopadhyay et al., 2000
).
Cetacean St values occurred in the present study most frequently in
the range 0.225<St<0.275 (Figs
5,
7AC). Peak Froude
efficiencies were found in this same range
(Fig. 6), which straddles the
lower boundary of the 0.250.35 St range predicted. Although
there are no cetacean or fish efficiency measurements to compare with, there
are relevant data for two-dimensional foils. Bandyopadhyay
(2002
) reported the peak
efficiency for a pair of tail flapping foils to be below the 0.250.35
St range. The efficiency for oscillating foils reported by
Triantafyllou et al. (1993
),
although maximal in the St range between 0.2 and 0.35, is practically
flat. The efficiency versus St data of Anderson et al.
(1998
;
Fig. 5) for a two-dimensional
foil clearly do not show peak efficiencies tuned to an St range
between 0.25 and 0.35. The lack of closer agreement with the present data is
not surprising given that the foil experiments could not (and were not assumed
to) capture the full flow field dynamics of swimming animals.
St is often expressed as the product of non-dimensional
peak-to-peak fluke amplitude (A/L) and non-dimensional
frequency [f/(U/L)]. Plotting the data in these
coordinates and including contours of constant St values provides an
additional perspective on the parameters controlling the range of St
encountered (Fig. 8). Regardless of St value, 89% of all the A/L data
(n=267) falls within a range of 0.150.25. Cetaceans are known
to maintain a nearly constant A/L of 0.20 with respect to
U (Kayan and Pyatetskiy,
1977; Fish, 1998
;
Fish et al., 2003
). A value of
0.20 for A/L is also typical for other animals that swim by
movements of the body and tail (Bainbridge,
1958
; Webb, 1975
;
Videler, 1993
;
Fish, 1998
;
Schultz and Webb, 2002
;
Fish et al., 2003
).
|
Swim speed is controlled through frequency modulation, as is common for
fish and other marine mammals that swim by oscillations of high aspect ratio
hydrofoils (Bainbridge, 1958;
Hunter and Zweifel, 1971
;
Webb and Kostecki, 1984
;
Feldkamp, 1987
;
Fish et al., 1988
;
Scharold et al., 1989
).
Regardless of St value, 90% of f/(U/L)
data falls within a range of 12. Note, if
A/L=0.150.25 and
f/(U/L)=12, the corresponding St
range is 0.150.50. Presumably the `boundaries' imposed by
A/L and f/(U/L) on St
are manifestations of additional morphological and hydrodynamic constraints
imposed on the animal. Optimal St values for pitching, heaving and
flapping foils have also been found to depend on other kinematic parameters,
including angle of attack, amplitude-to-chord ratio and phase of motion
(Anderson et al., 1998
;
Wang, 2000
;
Bandyopadhyay et al., 2000
;
Read et al., 2003
).
Representative St has often been estimated by simply multiplying
the slope of frequency versus speed over length by the mean
non-dimensional peak-to-peak fluke amplitude
(Triantafyllou et al., 1993;
Bandyopadhyay, 2002
). This
assumes the f versus U/L data passes through the origin. The
slope of f versus U/L throughout the present data is 0.89
(Fig. 2). The mean
A/L is 0.21. This product results in an estimate of 0.19 for
St for odontocete cetaceans. However, if the product of
f/(U/L) and A/L is first
calculated for each observation and then averaged, St=0.27. A 30%
disparity results because the y-intercept (0.58; see
Fig. 2) of the least squares
estimate of f versus U/L was initially ignored.
Whereas 91% of all the cetacean data fall between St=0.2 and 0.4,
a significant fraction (37%) of the D. leucas data conspicuously fell
outside this range. Mean St for D. leucas is
0.35±0.10 compared with 0.27±0.06 for the other cetaceans
examined in this study. It has been noted that the swimming performance and
efficiency of D. leucas differ considerably from those of other
cetaceans, which is consistent with its general body contour and low aspect
ratio flukes (Fish, 1998).
D. leucas generally exhibits the poorest swimming performance of
cetaceans. This species feeds on slow-moving prey, including crustaceans and
annelids (Brodie, 1989
). The
mean St for S. frontalis was also high at 0.33±0.03.
Unlike D. leucas, S. frontalis is a fast swimmer
(Fish and Rohr, 1999
). The
individual S. frontalis examined in this study had a mean
A/L of 0.25±0.02, which was the highest of any
species tested.
The data presented in this study significantly expand the previously used
Strouhal data for cetaceans (Triantafyllou et al.,
1991,
1993
;
Triantafyllou and Triantafyllou,
1995
; Taylor et al.,
2003
) by greatly increasing the number of observations (more than
100-fold), species (7-fold) and range of Reynolds number (10-fold) and, for
the first time, provide a direct comparison between measured Strouhal number
and swimming efficiency. The present data show that over the range of swim
speeds observed, 28 m s1, cetaceans swim at
St values between 0.2 and 0.4, preferring a range of 0.20.3
where maximum efficiencies occur. The strong relationship between maximum
propulsive efficiency and St continues to support the premise
(Triantafyllou et al., 1991
,
1993
) that vorticity control
associated with fluke kinematics is an important attribute of cetacean
swimming performance.
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Acknowledgments |
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References |
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