Hydrodynamic stability of swimming in ostraciid fishes: role of the carapace in the smooth trunkfish Lactophrys triqueter (Teleostei: Ostraciidae)
1 Department of Organismic Biology, Ecology, and Evolution, University of
California, Los Angeles, CA 91606, USA
2 Options of Bioengineering and Aeronautics, California Institute of
Technology, Pasadena, CA 91125, USA
3 Department of Aerospace Engineering, Technion, Haifa, 3200,
Israel
4 School of Natural Resources and Department of Biology, University of
Michigan, Ann Arbor, MI 48109, USA
* Author for correspondence (e-mail: ikbartol{at}lifesci.ucla.edu)
Accepted 11 November 2002
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Summary |
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Key words: boxfish, Lactophrys triqueter, stability, hydrodynamics, swimming, pressure, particle image velocimetry, moment
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Introduction |
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Powered correction systems are used by many fishes under a multitude of
conditions to regulate trajectories. For powered systems, neural processing is
required before the effectors make corrections and may ultimately be the
limiting factor for actively regulating trajectories
(Webb, 2000). Many
disturbances, especially those that are external, have low predictability in
terms of direction, period or amplitude. Under these conditions, accurate
phasing of correction forces with perturbations is difficult. Without proper
phasing, correction may even amplify the disturbance through `pilot-induced
error' (Webb, 1998
,
2000
). Trimming
self-correcting systems, on the other hand, can accommodate these
unpredictable events more effectively, since response latency periods are not
required. Consequently, fish relying more heavily on trimming control
seemingly are better adapted to more turbulent regimes than fish emphasizing
powered-control systems. Support for this hypothesis is provided by Webb
(1998
), who discovered that
the smallmouth bass Micropterus dolomieu, a fish that relies heavily
on powered control, has greater difficulty holding station in the wake of
cylinders than the river chub Nocomis micropogon, a fish that relies
heavily on self-correcting trimming control.
Whether using trimming or powered hydrodynamic systems, controlling
trajectories and avoiding growth of disturbances into energy-wasting
displacements is essential to organisms for effective and economical swimming.
However, surprisingly little is known about stability of locomotory motions in
fishes and other aquatic organisms, as noted by Blake
(1981), Weihs
(1993
,
2002
) and Webb
(2000
,
2002
). An intriguing fish that
resides in a highly turbulent environment and may take advantage of both
trimming and powered control is the smooth trunkfish Lactophrys
triqueter. The smooth trunkfish is a neotropical reef-dwelling fish that
has a rigid bony carapace consisting of hexagonal plates (or scutes), which
encases about 70-75% of its body (Tyler,
1980
). The carapace, a feature that distinguishes it from most
flexible-bodied marine fishes, is predominantly triangular in cross-section,
with one dorsal and two prominent ventro-lateral keels
(Fig. 1). In smooth trunkfishes
and other marine boxfishes (Teleostei: Ostraciidae), the carapace limits body
movements to locations posterior to the caudal peduncle. As a result, these
fishes rely heavily on complex combinations of movement of their five fins for
swimming.
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Field observations and recent studies on the swimming physiology of
boxfishes indicate that they are capable of remarkably low recoil motions,
both when moving in flumes and in highly variable velocity fields in nature,
resulting in smooth, energy-efficient, rectilinear swimming trajectories
(Gordon et al., 2000;
Hove et al., 2001
). Although
there are some studies describing and analyzing swimming in rigid-bodied
ostraciiform fishes (Blake,
1977
,
1981
,
1983a
,b
;
Hove et al., 2001
), little is
known about how this remarkable stability is achieved. The objective of our
study was to understand what role the uniquely shaped carapace of the smooth
trunkfish plays in maintaining this stability. We were particularly interested
in trimming self-correcting forces for pitching and yawing.
We approached this objective using three separate but interrelated techniques. For each approach, an anatomically exact, stereolithographic model of the smooth trunkfish was used. The three techniques applied were: (1) digital particle image velocimetry (DPIV), which provides a global picture of flow structures at various planes along the carapace; (2) pressure distribution measurements, which provide information on what is happening at the surface of the carapace, a region that is difficult to resolve using DPIV; and (3) force balance measurements, which provide an integrated view of forces acting on the carapace.
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Materials and methods |
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The specimen was scanned using a GE CT/i high-speed scanner (General Electric Medical Systems, Milwaukee, WI, USA). A continuous scan was performed through the entire fish using a 1.0 mm collimation at 120 kVp/200 mA-1 and a 1 s rotation time. A 50% overlap was used so that images were reconstructed every 0.5 mm. A total of 340 consecutive, two-dimensional (2-D), cross-sectional exposures were generated longitudinally from the snout to the posterior edge of the caudal fin. An appropriate threshold (contrast) was selected for maximum resolution of the carapace, and the images were converted to digital stereolithography (STL) files using a Marching Cubes algorithm, a high-resolution 3-D surface construction algorithm (Lorensen and Cline, 1987). Exact 3-D physical models of polymerized epoxy resin, one in halves and the other whole, were created from the files using stereolithographic rapid-prototyping (Solid Concepts, Inc., Valencia, CA, USA).
Measurements of CT images
A variety of morphological measurements were collected from the 2-D,
cross-sectional CT images to characterize the unique shape of the carapace.
Before measurements were taken, the CT images were converted from a UCLA PACS
in-house format to a TIFF format. Measurements were collected from every
fourth image using NIH Image v. 1.61 (National Institute of Health,
http://rsb.info.nih.gov/nih-image/).
For each CT image considered, carapace width (distance from ventro-lateral
keel tip to ventro-lateral keel tip), carapace height (distance from dorsal
keel tip to midline of ventral region), eye ridge width (distance from eye
ridge tip to edge ridge tip), dorsal keel angle (determined using the apex of
the keel (vertex) and the two widest points on either side of keel apex), and
both ventro-lateral keel angles (determined similarly to dorsal keel angles)
were measured along the length of the carapace
(Fig. 1). To characterize
levels of lateral and ventral concavity/convexity, a segment connecting the
apex of each ventro-lateral keel and the apex of the dorsal keel
(SL) and a segment connecting the apices of the
ventro-lateral keels (SV), respectively, were drawn on all
CT images considered using NIH image. The maximum distance(s) the body
indented and/or extended relative to the segment and location(s) of maximum
indentation/extension along the segment were recorded.
Digital particle image velocimetry
Willert and Gharib (1991)
and Raffel et al. (1998
)
provide detailed descriptions of the DPIV technique for flow field
measurements. A general description of the experimental techniques relevant to
our study is given here. To reduce glare from laser light, one of the two
fabricated models was painted black, which did not obscure the hexagonal scute
topography at the surface of the carapace. For DPIV measurements made along
the body, the tail of the model was removed and replaced with a 10 cm rod
(also painted black and 1.0 cm in diameter, which was similar in dimension to
the caudal peduncle). The rod was connected to a sting that entered the water
tunnel from above. For DPIV measurements in the wake, the tail was reattached,
and the sting was affixed to a machined connector located at the caudal
peduncle. We used a water tunnel with a 30 cmx30 cmx100 cm test
section (Model 503, Engineering Laboratory Design, Inc., Lake City, MN, USA)
seeded with silver-coated hollow glass spheres (14 µm in diameter). Two
pulsed ND:YAG lasers (wavelength=532 nm, power rating 50 mJ; New Wave
Research, Fremont, CA, USA), a series of front-surface mirrors, and a
cylindrical lens were used to generate and align an illuminated sheet
approximately 1.0 mm thick. The laser sheet was projected beneath the water
tunnel in a transverse (YZ) plane. A Pulnix CCD video camera
(TM-9701) with a frame size of 480x768 pixels and frame rate of 30 Hz
was positioned downstream of the working section (unobstructed views of
oncoming flow were made possible through a Plexiglas exit tank).
The video camera, lasers and a Coreco OC-TC10-DIGSE frame grabber (National
Instruments, Inc., Austin, TX, USA) were synchronized using a video timing box
and FlowVisionTM software (PixelFlow, Pasadena, CA, USA). The
video camera generated timing signals for phase-locking the frame grabber,
lasers and camera. Generally, the two lasers were pulsed for 0.02 ms and
separated by a period of 1 ms during paired image collection. At least 60
sequential images (30 paired images) were considered for processing for each
trial. In all images except those collected in the wake, the model was
subtracted out of the video image prior to processing. The interrogation
window (i.e. the area over which the particle shifts were averaged) was
322pixels with a 16 pixel offset (50% overlap). The displacement of
particles within the interrogation windows comprising the paired images was
calculated using cross-correlation
(Willert and Gharib, 1991).
Outliers, defined as particle shifts that were 3 or more pixels greater than
particle shifts of neighboring particles, were removed and corrected, and data
were smoothed to remove high frequency fluctuations within the video
images.
The resulting flow displacement fields were divided by the time difference between the paired images to determine the velocity field of the flow. Mean velocity and mean vorticity fields were determined for each trial using FlowVisionTM software. Circulation of regions of concentrated vorticity attached to the model was determined by carefully tracing along the outline of the model where the region of concentrated vorticity was present, tracing along the remaining vorticity contour found external to the model, and then computing the area integral within the tracing using an in-house code at the California Institute of Technology. Circulation of vortices in the wake was determined by tracing along the outlines of each region of concentrated vorticity and computing the area integral of the tracing.
DPIV measurements were collected at transverse planes along the length and
within the wake of the model as it was positioned at various pitching angles
of attack in the water tunnel. The water tunnel was set at a speed of 44 cm
s-1 (2.6 BL s-1), where BL=body
length, which corresponds to roughly half the sustained speed range of boxfish
(J. Hove, unpublished swim tunnel data). The water tunnel was also set at low
speeds (10 cm s-1, 0.6 BL s-1) and high speeds
(90 cm s-1, 5.3 BL s-1) periodically to confirm
that DPIV results at the intermediate speed were qualitatively applicable over
the range of swimming speeds of the fish. Data were collected at 2°
intervals between -10 and +10°, and at 5° intervals at more negative
and positive angles of attack, up to -30 and +30°, respectively. At all
pitching angles of attack, flow patterns were examined in the wake of the
model (5 cm downstream from the caudal fin) and at five locations along the
body of the model: the eye ridge, the point of maximum girth, the midpoint
between the point of maximum girth and the posterior edge of the carapace, the
posterior edge of the carapace and the caudal peduncle. Several (2-3) trials
were performed at each location at each pitching angle of attack. The model
was oriented sideways, upside-down, and right-side-up in the water tunnel
during the various angles of attack to ensure that no zones around the
carapace were ignored because of shadowing. The upside-down orientation worked
best for positive pitching angles, while the sideways orientation worked best
for negative pitching angles. Lift on the carapace was computed at each angle
of attack considered using a 2-D airfoil lift equation
(Tritton, 1998):
Ls=-
wu
, where Ls
is lift per unit span (kg s-2),
w is freshwater density (kg
m-3), u is flow speed (m s-1), and
is
mean circulation of each vortex comprising the pair in the wake of the model
(m2 s-1). These vortices represent both the linear and
non-linear contributions to slender wing-body lift
(Küchemann, 1978
). To
obtain total lift L, Ls was multiplied by the
span of the ventral region of the boxfish model.
Additional DPIV studies were performed at transverse planes along the length of the model as it was positioned at various yawing angles of attack (0 to +30°) in the water tunnel set at the same speed described above. Data were collected at yawing angles of attack of 0, 10, 20 and 30° at the following locations: the eye ridge, the point of maximum girth, the midpoint between the point of maximum girth and the posterior edge of the carapace, the posterior edge of the carapace, and caudal peduncle. Several (2-3) trials were performed at each location at each yawing angle of attack. As was the case for pitching studies, the models were positioned in different orientations to ensure that no shadowed region was overlooked. The upside-down orientation worked best for yawing studies.
Pressure measurements
The second smooth trunkfish model, constructed in halves but identical in
dimensions to that employed in DPIV experiments, was used for pressure
experiments. One of the model halves was hollowed out, and 36 holes were
drilled in lateral and ventral regions of the carapace. The holes were drilled
along body transects similar to those examined in the DPIV studies. Urethane
tubing (i.d. 0.068 cm, o.d. 0.129 cm) was inserted into the holes and glued in
place so that the tubing was flush with the model surface. The two model
halves were glued together. Tubing exited the model through a 1.0 cm diameter,
10 cm long rod attached to the posterior section of the model. The rod was
used to mount the model to a sting in a 61 cm wind tunnel (model 407,
Engineering Laboratory Design, Inc., Lake City, MN, USA). Tubing exiting the
model was connected to a Scanivalve 48-channel rotating pressure scanner
(Scanivalve Inc., Liberty Lake, WA, USA) and a barocel electric manometer and
capacitative differential pressure sensor (Barocel Datametrics, Wilmington,
MA, USA). Static pressure (N m-2) at each of the 36 ports was
expressed relative to static pressure (N m-2) at a tunnel wall
port.
Data were collected while each model was positioned at 2° intervals for
pitching angles of attack of -30 to +30° and yawing angles of attack of 0
to 30° (for yawing angles of attack the pressure ports were both exposed
to flow (i.e. positioned in the near field carapace side) and shielded from
flow (i.e. positioned in the far field carapace side) for each of the angles
of attack). For the angles considered, data were acquired at 100 Hz for 10 s
using LabVIEW software (National Instruments, Inc.). Wind tunnel speed was set
according to the Reynolds number considered in water tunnel trials. Pressure
coefficients (CP) were calculated by dividing the pressure
difference above (N m-2) by dynamic pressure
[AU2/2; where
A is air density (kg
m-3) and U is wind speed (m s-1)], determined
using a Pitot-static tube positioned in the wind tunnel.
Force measurements
The smooth trunkfish model used in DPIV experiments was also used in force
measurement experiments. The model was mounted caudally to a sting in a water
tunnel with a test section 61 cmx46 cmx244 cm in dimension. Force
measurements were collected using three Interface 2.27 kg strain gauge load
cells (Interface, Inc. Scottsdale, AZ, USA) [two load cells measured forces
normal to flow (lift), one load cell measured forces parallel to flow (drag)]
connected to an in-house force balance
(Lisoski, 1993). Data output
from the load cells were amplified using three Interface SGA
amplifiers/conditioners and recorded with a Dash 8 Series data recorder
(Astro-Med, Inc.). Data were collected at 200 Hz for 10 s for each angle
considered. As was the case for pressure experiments, force data were
collected every 2° from pitching angles of attack of -30 to +30° (some
additional angles <-30° and >+30° were also considered). Flow
speed during trials was identical to that considered in DPIV experiments [44
cm s-1 (2.6 BL s-1)].
Coefficients of drag (CD) and lift
(CL) were calculated using steady-state equations
[CD=2xD/(wxAfxu2)
or
CL=2xL/(
wxApxu2),
where D is total drag (N), L is total lift (N),
w is
water density (kg m-3), Af is maximum frontal
area (m2), Ap is planform surface area of the
ventral region of the carapace (m2), and u is water tunnel
speed (m s-1)]. Pitching moments (M) about the center of
mass of the models were computed using the equation:
M=d1(L2-L1)+(L1+L2)x(d2cos
)+D(d2sin
),
where d1 is distance (m) between load cell beams in the
force balance, d2 is distance (m) between center of force
balance and center of mass of model, L1 is lift (N)
measured at load cell closest to model, L2 is lift (N)
measured at load cell farthest from model, D is drag (N) on the
model, and
is angle of attack (degrees) of the model. Pitching moment
coefficients (CM) were computed using the equation:
CM=2M/(
wApcU2),
where c is the chord length of carapace (m).
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Results |
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Significant regions of convexity were present laterally along the carapace, especially in areas just posterior to the eye (25-40% CL) and near the edge of the carapace (>84% CL) (Fig. 3). Along a given dorso-ventral transect, maximum lateral convexity occurred 0.84-2.85 cm below the dorsal keel (Figs 1, 3). Ventral convexity, which was always greatest equidistant from the ventro-lateral keels along transverse transects, actually decreased in magnitude posteriorly until 71% CL, at and after which no significant convexity was present (Figs 1, 3). Lateral concavity was present in locations from 38-92% CL. Along dorso-ventral transects, maximum lateral concavity occurred 0.38-1.23 cm above the ventro-lateral keels (Figs 1, 3). Ventral concavity was also present in locations from 58 to 94% CL. Along dorso-ventral transects, maximum ventral concavity occurred 0.62-1.37 cm from the ventro-lateral keels (Figs 1, 3).
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Digital particle image velocimetry
At positive pitching angles of attack, regions of concentrated, discrete
vorticity began to develop near the anterior edges of the ventro-lateral keels
at a longitudinal location corresponding to the eye ridge
(Fig. 4). These regions of
concentrated vorticity intensified posteriorly along the ventro-lateral keels,
both in terms of peak vorticity and circulation, until two well-developed,
counter-rotating vortices ultimately formed at the posterior edge of the
carapace. The vortices left the body completely at the caudal peduncle. Some
regions of concentrated, attached dorsal vorticity formed along the body as
well, especially around the eye ridge, where vortex circulation was stronger
dorsally than ventrally at positive angles of attack <20°. However,
peak dorsal vorticity and dorsal circulation did not grow steadily along the
body, and did not reach the circulation magnitudes observed in ventral areas
at the posterior edge of the carapace and at the caudal peduncle. In the wake,
dorsal and ventral vortices merged with the vorticity representing the linear
lift, producing two well-developed, counter-rotating vortices with a prominent
downwash of flow between them (Fig.
4).
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At negative pitching angles of attack, regions of concentrated, attached vorticity also began to develop around the ventro-lateral keels at a longitudinal location corresponding to the eye ridge (Fig. 5). As was the case for positive angles of attack, circulation and peak vorticity of ventral vortices increased posteriorly, developing into two counter-rotating vortices at the posterior edge of the carapace, and ultimately leaving the body completely at the caudal peduncle. However, while regions of concentrated vorticity developed above the ventro-lateral keels at positive pitching angles of attack, regions of concentrated vorticity formed below the ventro-lateral keels at negative pitching angles of attack (Figs 4, 5). Furthermore, vortical flow rotation changed, depending on angle of attack. When viewed from the rear, flow around the right ventro-lateral keel was counterclockwise at positive angles of attack and clockwise at negative angles of attack (Fig. 6). Some regions of weaker, concentrated, attached vorticity formed around the eye ridge in dorsal locations at negative angles of attack, but as was the case at positive angles of attack, the dorsal vortices did not grow steadily along the body like the ventral vortices. In the wake, two well-developed vortices were present with a prominent upwash of flow between them (Fig. 5).
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At pitching angles of attack close to 0°, the same general patterns of vortex development occurred: regions of concentrated, attached vorticity formed at the anterior edges of the ventro-lateral keels and grew in terms of peak vorticity and circulation posteriorly, while weaker regions of dorsal vorticity formed sporadically with no consistent antero-posterior circulation growth (Fig. 7). However, circulation and peak vorticity of ventral regions of concentrated, attached vorticity were lower at angles of attack near 0° than at more positive or more negative angles of attack (Figs 6, 7). Irrespective of angle of attack, circulation of attached ventral vortices was always greatest at the posterior edge of the carapace. Interestingly, at an angle of attack of 0°, vortical flow rotation and position resembled conditions at positive angles of attack.
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When the carapace was positioned at yawing angles of attack, vortex development occurred adjacent to far-field regions of the carapace (i.e. portions of the carapace that are shielded somewhat from oncoming flow), but little concentrated vorticity formed in near-field locations of the carapace (i.e. portions of the carapace that are exposed to oncoming flow) (Fig. 8). At the far-field side of the carapace (the far-field side in this case is the right side of the carapace when viewed from the rear), a clockwise region of concentrated, attached vorticity developed around the eye ridge/anterior edge of the dorsal keel and grew posteriorly in terms of circulation and peak vorticity along the keel. The resulting vortex left the body at either the posterior edge of the carapace or caudal peduncle. One or two counterclockwise regions of concentrated, attached vorticity developed at the anterior edge of the ventro-lateral keel and grew posteriorly along the keel, before leaving the body at either the posterior edge of the carapace or the caudal peduncle. As yawing angles of attack deviated farther from 0°, peak vorticity and circulation of far-field vortices intensified (Fig. 8). As was the case with pitching angles of attack, peak vorticity and circulation of attached vortices were always greatest at the posterior edge of the carapace, irrespective of angle of attack.
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Pressure measurements
Locations of the concentrated, attached vorticity observed in DPIV
experiments were closely correlated with locations of low pressure detected in
pressure experiments. Along the eye ridge dorso-ventral transect, regions of
attached vorticity developed around the eye ridge and above the ventro-lateral
keel at positive pitching angles of attack. At these locations along the
transect, which correspond to pressure ports B4 and B1, respectively, low
pressure was also detected (Figs
4,
9). Along the maximum girth
dorso-ventral transect, regions of concentrated vorticity were detected
laterally and above the ventro-lateral keel, which correspond to pressure
ports B7 and B11, respectively. Low pressure at B11 was clearly apparent at
all positive pitching angles of attack along the transect. Low pressure at B7
was less apparent, but there was a drop in pressure at B7 relative to its
dorsal neighbor B6 at angles of attack >4°
(Fig. 9). Along a transect half
way between the point of maximum girth and the posterior edge of the carapace,
both attached vorticity and low pressure were observed just above the
ventro-lateral keel (pressure port B12) at positive pitching angles of attack.
At the posterior edge of the carapace, a strong vortex was observed above the
ventro-lateral keel at positive pitching angles of attack; at this location
(B20 on pressure model) low pressure relative to other points along the
dorsoventral transect was observed. Furthermore, pressure dropped at ports
near regions of concentrated, attached vorticity as angles of attack
increased, which is consistent with the observed increase in peak vorticity
and circulation with higher angles of attack
(Fig. 9).
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Along the eye ridge dorso-ventral transect, regions of attached vorticity formed around the ventro-lateral keel and the eye ridge at negative pitching angles of attack. Consistent with this observed vorticity pattern, a clear decrease in pressure was observed at B1, a pressure port just above the ventro-lateral keel (Fig. 10). As angles of attack increased, pressure dropped at B1, just as vortex circulation increased at more negative angles of attack. A drop in pressure was also observed at pressure ports near the eye ridge; for angles of attack 0 to -10°, pressure was low at B4, and for angles of attack -20 to -30°, pressure was low at B5. For dorso-ventral transects located at maximum girth and at more posterior locations, regions of concentrated vorticity were located largely just below the ventro-lateral keel at negative angles of attack, an area where no pressure port was located. As a result, there were no consistent declines in pressure at ports just above the ventro-lateral keel (i.e. B11, B12 or B20) along dorso-ventral transects as observed at positive angles of attack. Interestingly, at the point of maximum girth, there was a noticeable drop in pressure at A4, a pressure port located in the middle of the ventral region of the carapace that presumably was influenced by regions of ventrally located, concentrated vorticity (Figs 5, 10).
|
Although peak vorticity and vortex circulation increased consistently along the ventro-lateral keels from the anterior edges of the keels to the posterior edge of the carapace, pressure values did not decrease antero-posteriorly at positive pitching angles of attack. Instead, pressure just above the ventro-lateral keel decreased from the snout to the point of maximum girth (B11), but then increased posteriorly thereafter (Fig. 11). Pressure along the antero-posterior path followed by regions of concentrated ventral vorticity at negative angles of attack was not measured because of a lack of pressure ports. However, along the D1B20 antero-posterior transect considered for positive angles of attack, pressure was lowest at B1, the only location along the transect where regions of concentrated vorticity were prominent. Along an antero-posterior transect where regions of attached dorsal vorticity were frequently observed, pressure was lowest above and/or behind the eye ridge (D4, B5) at both positive and negative angles of attack (Fig. 11), areas where regions of concentrated, attached dorsal vorticity were frequently greatest. Dorsal pressures at positive angles of attack were lower than at negative angles of attack, which too is consistent with the observed vorticity patterns, i.e. there was stronger concentrated vorticity around the eye ridge at positive as opposed to negative angles of attack.
|
At yawing angles of attack, vortex formation occurred at the far-field side of the carapace, most prominently in regions adjacent to the dorsal and ventro-lateral keels. Low pressure was consistently detected in these regions along dorso-ventral transects. For example, at the point of maximum girth, pressure at D7 (port just below dorsal keel) and B11 (port just above ventro-lateral keel) was low when models were positioned at yawing angles of attack with the ports in the far field (Fig. 12). Moreover, a clear drop in pressure at lateral locations between the dorsal and ventro-lateral keel (i.e. B7-B9) was frequently observed, which is where additional regions of concentrated vorticity were detected using DPIV (Figs 8, 12). As circulation and peak vorticity increased with increasing yawing angle of attack, pressure in regions of concentrated vorticity dropped. When the model was positioned so that the holes were in the near field, i.e. directly exposed to flows, a very different pressure distribution occurred. As the pressure ports were exposed to a larger normal velocity component in the near-field side (i.e. higher angles), pressure actually rose at all the lateral ports (Fig. 12).
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Force balance measurements
No obvious stall occurred at angles of attack up to ±30°, and
overall lift coefficients of the carapace were similar to coefficients of
delta wings of similar aspect ratio (Fig.
13). The lift coefficient of the carapace was closest to 0 at
-2° (CL=-0.005). At 0°, CL was
0.027. The drag coefficient of the carapace was lowest at -4°
(CD=0.157) and was 0.167 at 0°. At positive angles of
attack, the lift-to-drag ratio was highest at +20°, which is consistent
with delta wings that often have maximum L/D ratios >15°
(Bertin and Smith, 1989). At
negative angles of attack, the highest downward-directed lift to drag ratio
occurred at -16°. Nose-down pitching moments about the center of mass were
detected at angles of attack >-2°, while nose-up pitching moments about
the center of mass were detected at angles of attack <-2°
(Fig. 13).
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When compared to lift measurements from force balance experiments, lift values predicted from DPIV circulation values (using 2-D airfoil equations) were consistently of higher magnitude (paired sample t-test, mean difference=0.045, d.f.=6, t-value=4.322, P=0.005). At positive angles of attack, DPIV lift estimates were 29-33% higher in the positive direction than force balance lift measurements. At negative angles of attack, DPIV lift estimates were 25-38% higher in the negative direction than force balance lift measurements (Fig. 14).
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Discussion |
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Morphological features
The results of this study indicate that prominent vortices form along the
body when smooth trunkfish L. triqueter pitch and yaw. The
morphological features of the carapace play an instrumental role in the
generation, growth and persistence of these vortices. The anterior origins of
the ventro-lateral keels, which were present at locations approximately 14%
CL, force flow separation and the generation of vorticity during
pitching and yawing. The ventro-lateral keels extend along the majority of the
length of the carapace, becoming sharper posteriorly up to a location 82%
CL and providing a large area for vorticity buildup along the body.
These characteristics facilitate vortex circulation growth in areas adjacent
to the ventro-lateral keels. Regions of lateral concavity found 38-92%
CL and 0.38-1.23 cm above the ventro-lateral keels and ventral
concavity present 58-94% CL and 0.62-1.37 cm below the ventro-lateral
keels provide channels for vortices to develop, and serve to fix developing
vortices in place at various pitching and yawing angles of attack. The steep
lateral sides and prominent dorsal ridge also play important roles in
isolating developing ventro-lateral flows, allowing counter-rotating vortices
to form without interaction with one another. The eye ridge, which was
consistent in width and present at 14-37% CL, and dorsal keel, which
became sharper posteriorly up until approximately 70% CL, are
responsible for forcing flow separation and the generation of vorticity in
dorsal regions when smooth trunkfish pitch and yaw, respectively. During
yawing, vortex circulation growth along the carapace is facilitated by the
dorsal keel when one side of the keel is exposed to larger normal velocity
components than the other. However, during pitching, the lack of multiple,
laterally extended dorsal keels, which would allow the two regions of
vorticity shed from the eye ridge to grow in strength posteriorly with minimal
interaction with one another (as in ventral regions), inhibits vortex
circulation growth along the carapace. Consequently, vortex circulation in
posterior regions of the carapace is weaker dorsally than ventrally during
pitching.
Self-correction for pitching
DPIV measurements, which provide a global picture of flow around the body
within planar domains, indicate that pitching angle of attack has a large
impact on ventral vortex formation. As angles of attack increased from
approximately 0° in the positive direction, vortices with stronger peak
vorticity and circulation developed along concavities above ventro-lateral
keels, beginning at the anterior origins of the ventro-lateral keels and
reaching maximum strength at postero-lateral regions of the carapace. As
angles of attack increased from approximatey 0° in the negative direction,
vortices with stronger peak vorticity and circulation developed along ventral
concavities below the ventro-lateral keels, beginning again at the anterior
origins of the ventro-lateral keels and reaching maximum strength at
postero-ventral regions of the carapace. At angles near 0°, weak vortices
with the lowest circulation developed along the ventro-lateral keels.
Pressure measurements, which provide useful information on flow conditions
at the surface of the carapace, an area difficult to resolve with DPIV because
large velocity gradients within the boundary layer require high particle
densities and small interrogations windows to resolve, were consistent with
the global flow patterns. In regions where attached, concentrated vorticity
was observed, areas of low pressure were consistently detected on the carapace
surface. Moreover, as angles of attack deviated farther from 0°, either in
a positive or negative direction, and circulation of attached vortices
increased, pressure dropped accordingly. These correlations are in agreement
with Bernoulli's Law, which states that higher local speeds result in lower
static pressure. Near the core of each vortex, flow speeds are higher than the
surrounding fluid and consequently static pressure is low. As the vortex
intensifies in strength and local speeds increase, pressure becomes more
negative (McCormack and Crane,
1973; Tritton,
1998
; Vogel,
1994
). Consequently, for those ports located near the vortex core,
a low-pressure region is detected and becomes more conspicuous as angles of
attack increase. Along some dorso-ventral transects, a high-pressure region
adjacent to the low-pressure vortex core region was present (e.g. B2 and B10
in Fig. 9). This high-pressure
region is indicative of a slow flow area where separated flow comes to
reattachment, a process that is visible in the velocity vector and vorticity
contour fields for the respective transects
(Fig. 4). The farther the
vortex core migrates from the carapace surface, the less effect it has on
pressure at the surface because the induced velocity drops with growing
distance. This point is illustrated by the observed increase in pressure along
an antero-posterior transect above the ventro-lateral keel, beginning at
maximum girth and progressing posteriorly to the edge of the carapace. Despite
an observed increase in vortex circulation along the antero-posterior
transect, localized surface pressures did not continue to decrease along the
transect because the vortex core migrated away from the carapace, thus having
less effect on surface pressure (see Figs
4,
11). This result highlights
the importance of using pressure and DPIV data in conjunction; linking global
flow features with their concomitant localized effects on the smooth trunkfish
carapace is critical for fully understanding flow effects.
Force measurements, which afford an integrative view of the forces acting
on the entire carapace, provide further support for the flow patterns detected
in DPIV and pressure experiments. Based on DPIV and pressure results, vortices
are generated near the anterior of the fish and grow in strength as they move
posteriorly down the carapace, most prominently in regions adjacent to
ventro-lateral keels. Delta wings, which have comparable planforms to smooth
trunkfish, produce similar flows. In delta wings, a coiled vortex sheet with a
core of high vorticity forms at the leading edge of the wing and grows
posteriorly along the wing generating lift a process that differs from
lift created through bound circulation in conventional wings and leads to
higher angles of attack for stall (Bertin
and Smith, 1989). The observed similarity in lift coefficients and
L/D ratios between delta wings and smooth trunkfish is thus further evidence
of vortex generation and subsequent growth along the ventro-lateral keels.
Based on force measurements, lift coefficients were closest to 0 at -2°
and were slightly positive at 0°. This too is in agreement with DPIV data.
DPIV results indicate that lowest vortex circulation occurs at angles of
attack near 0°, and vortices are generated above ventro-lateral keels at
0°, providing beneficial lift for counteracting negative buoyancy present
in rigid-bodied ostraciids (Blake,
1977
).
The vortical flow patterns summarized here produce trimming forces that
self-correct for pitching motions, i.e. rotation in the vertical, head up/down
longitudinal plane. Attached vortices with the strongest peak vorticity and
circulation generally develop posterior to the center of mass; the center of
mass of smooth trunkfish is located at a longitudinal location corresponding
approximately to the point of maximum girth
(Bartol et al., in press).
These vortices form above (positive angles of attack) or below (negative
angles of attack) ventro-lateral keels that extend laterally at an angle of
25-40° relative to a horizontal axis when viewed in cross section.
Consequently, suction derived from the presence of a vortex above or below the
ventro-lateral keels, which was evident as low pressure zones in pressure
experiments, should act largely upward and posterior to the center of mass at
positive angles of attack (which also occurs in delta wings) and downward and
posterior to the center of mass at negative angles of attack. Based on
pitching moments recorded in force balance measurements, where nose-down
pitching moments occurred and became progressively stronger as angles of
attack became more positive, and nose-up pitching moments occurred and became
progressively stronger as angles of attack became more negative, this is
exactly what happens. Therefore, the ventro-lateral keels are effectively
generating self-correcting forces for pitching motions; the self-correcting
couple is proportional to the angle to which the fish is perturbed from a
horizontal swimming trajectory.
Dorsal and antero-posterior flows
Regions of attached, concentrated vorticity detected around the dorsal eye
ridge in this study did not grow in a uniform manner antero-posteriorly during
pitching, as was the case for ventral keel induced vorticity. Instead regions
of concentrated dorsal vorticity largely broke down along the body posterior
to the eye ridge, irrespective of pitching angle of attack. These results
again are consistent with the observed pressure distributions. If dorsal
carapace morphology does not facilitate the growth of vortices posteriorly, is
there an advantage to producing vorticity and creating the observed pressure
distributions at the eye ridge, i.e. is there a hydrodynamic purpose of the
eye ridge? Given that ambient pressure occurs around the eye ridge regardless
of angle of attack, the eye ridge may generate the observed flows to ensure an
optimal pressure distribution around the eye. Maintaining ambient pressure
around the eye is advantageous because the eye (and most importantly the lens)
will not be pushed in or pulled out as flow moves along the body, which
improves eye function. Detection of ambient pressures around the eyes has been
reported in other nekton, such as squid, bluefish, and tuna
(Aleyev, 1977;
Dubois et al., 1974
;
Vogel, 1987
). In addition to
creating an optimal pressure distribution around the eye, the eye ridge may
generate some lift to counteract the nose-down pitching moment produced by the
ventro-lateral keels at a 0° angle of attack, thus allowing for more
uniform lift generation about the center of mass to counteract negative
buoyancy. This proposed function is analogous to the role of canards on delta
wing aircraft, which provide nose-up trimming moment to counteract nose-down
pitching (Bertin and Smith,
1989
).
Based on pressure distributions along antero-posterior transects, there is
no appreciable flow separation from the posterior edge of the carapace in
ventro-lateral regions (i.e. pressure coefficients are not greater than
ambient pressure (CP=0) and pressure gradients are gradual). Flow
remains bound to the carapace, presumably because of the ventral vortices,
which pick up vorticity and energize flow close to the carapace surface. Based
on the observed positive pressure coefficients at the posterior edge of the
carapace in dorsal locations, flow separation appears to occur in dorsal,
posterior regions of the carapace for angles of attack >4°. According
to pressure distribution measurements of other aquatic organisms, flow
separation occurs near the eye and anterior to the arms of squid Loligo
pealei swimming tail-first (Vogel,
1987), at the posterior quarter of the body of penguins
Pygoscelis papua, P. aseliae and P. antartica and tuna
Trachurus mediterraneus (Aleyev,
1977
; Bannasch,
2000
), and at the posterior half of the body of a swordfish
Xiphias gladius (Aleyev,
1977
). Using surgically implanted catheters, Dubois et al.
(1974
) found that flow does
not separate anterior to the caudal peduncle in bluefish Pomatomus
saltatrix. Results from both Dubois et al.
(1974
) and the present study
are significant because they suggest that fin motion may not be necessary to
keep flow attached to the body, as suggested in other studies
(Aleyev, 1977
;
Blake, 1983b
).
Self-correction for yawing
The flow patterns and pressure distributions detected around and along the
carapace at various yawing angles of attack indicate that the carapace also
generates self-correcting forces for yawing, i.e. rotations in the left/right
horizontal frontal plane. Dorsal and ventral vortices clearly formed on the
farfield side of carapace, especially in areas adjacent to the dorsal and
ventro-lateral keels, when the smooth trunkfish model was positioned at
various yawing angles of attack. Circulation of attached vortices increased
posteriorly along the carapace, such that maximum vortex circulation occurred
posterior to the center of mass. Vortex circulation and peak vorticity also
increased as yawing angles increased. In areas where concentrated attached
vorticity was observed, low pressure was detected, and as circulation of
attached vortices increased with increasing angle of attack, pressure dropped
accordingly. Suction derived from the presence of vortices at far-field
locations of the carapace acts largely opposite the direction of the yaw and
posterior to the center of mass, thus providing trimming forces that
self-correct for yawing motions. As with pitching, the self-correcting couple
is proportional to the angle to which the fish is perturbed.
Hydrodynamic stability in boxfishes
Hove et al. (2001) found
that boxfishes exhibit some of the smallest amplitude recoil moments known
among fishes. As a result, they swim in smoother trajectories than either
body- and caudal-fin (BCF) or single-complex median- and paired-fin (MPF)
swimmers. Results from the three independent but complementary experimental
approaches applied in this study indicate that the keeled bony carapace plays
an important role in producing both longitudinal and latitudinal stability in
at least one species of boxfish, the smooth trunkfish L. triqueter.
In addition to producing self-correcting forces, the bony carapace of the
smooth trunkfish also appears to incur high drag; the drag coefficients
reported in this study are higher than those typically recorded in other fish,
by as much as an order of magnitude (Blake,
1981
,
1983a
). Given that Anderson et
al. (2001
) found that live,
flexible-bodied swimming fish have much greater friction drag than has been
reported in the literature because of boundary layer thinning and streamwise
acceleration of the near-field, drag differences between boxfish and
flexible-bodied fish may be less pronounced than expected. Nonetheless,
control for pitching and yawing is presumably more important than drag
reduction for fishes like smooth trunkfish. These fishes generally move
relatively slowly, but live in highly energetic waters with frequent external
disturbances, like turbulence, that can lead to large displacements and
significant energy-wasting trajectories. The trimming self-corrective forces
produced by the carapace limit these displacements and are especially
advantageous for the unpredictable velocity fields experienced by smooth
trunkfish because rapid neural processing a factor that may limit
correction responses to sudden, erratic perturbations is not required,
as in powered control systems (Webb,
2000
). In addition to the energy savings associated with
self-correction for disturbances, maintenance of smooth swimming trajectories
also presumably improves sensory acuity of both hostile and target objects
because it reduces complexity of movement, a factor that improves sensory
perception in other animals (Land,
1999
; Kramer and McLaughlin,
2001
).
Trimming and powered forces provided by the fins also presumably play an
important role in the control and regulation of trajectories. Like spotted
boxfish, smooth trunkfish have four low aspect-ratio median and paired fins of
nearly identical area that assuredly move in complex ways together with the
caudal fin to minimize recoil motions
(Hove et al., 2001; I. Bartol,
unpublished morphological measurements of smooth trunkfish). The pectoral
fins, in particular, may play an important role in the structure of vortices
forming along the ventro-lateral keels since the pectoral fins are located
near the path of body-induced vortex formation. Studies focusing on the
interactions between flows over the fins and over the bodies of live boxfishes
are underway in our laboratories. Although fin flow and body flow interactions
are present in live fishes, the results of this study are applicable to freely
swimming fishes because the underlying physical functioning of the carapace
self-correcting system is not affected by fin/body flow interactions. When
angles of attack deviate from 0° in either the positive or negative
direction, vortex strength increases along the carapace. The self-correcting
effect of these vortices will still occur, irrespective of whether vortex
strength increases relative to magnitudes measured for the body alone, or for
some fin-modified value.
Quantifying and visualizing flow patterns around oscillating fins poses a
significant challenge for understanding stability in smooth trunkfish. DPIV is
a powerful tool that has been applied to a number of interesting areas of fish
locomotion, ranging from wakes around flexing bodies
(Stamhuis and Videler, 1995;
Wolfgang et al., 1999
;
Müller et al., 2001
), fin
function (Drucker and Lauder,
1999
,
2000
,
2001a
; Wilga and Lauder,
1999
,
2000
), fluid forces during
turning maneuvers (Wolfgang et al.,
1999
; Drucker and Lauder,
2001b
), flows around the caudal peduncle and finlets
(Nauen and Lauder, 2001
),
vorticity control in fish-like propulsion
(Beal et al., 2001
), and
boundary layer flows (Anderson et al.,
2001
). However, DPIV has one significant limitation: it is a 2-D
measurement platform that does not consider all three flow-field components,
which may ultimately lead to some error when calculating propulsive and
resistive forces. An example of such error was encountered in the present
study. Lift forces determined using DPIV were consistently higher than those
measured directly using a force balance. A comparable situation was also
detected by Noca et al. (1997
)
when measuring instantaneous forces on a cylinder in an incompressible
cross-flow. These errors occurred because a 2-D rather than a 3-D velocity
field was used to determine force.
Future studies on live smooth trunkfish will be performed using defocusing
digital particle image velocimetry (DDPIV), a new method that allows us to
visualize and quantify flows in three dimensions as they move along the body
and around the fins (Pereira et al.,
2000; Pereira and Gharib,
2002
). Since this technique allows for the quantification of flows
over a significant volume of fluid, it is an improvement on, not only current
2-D DPIV systems, but also stereo-DPIV technology, where 3-D flows are imaged
in a thin slice of fluid. Employing DDPIV technology to study stability in
live rigid-bodied boxfishes promises to provide unprecedented information on
the control and regulation of trajectories in rigid-bodied multi-propulsor
swimmers.
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Acknowledgments |
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