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First published online September 19, 2008
Journal of Experimental Biology 211, 3147-3159 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.020263
Pitching stabilization via caudal fin-wave propagation in a forward-sinking parrot cichlid (Cichlasoma citrinellum x Cichlasoma synspilum)
1 Department of Power Mechanical Engineering, National Tsing Hua University,
Hsinchu, Taiwan
2 Department of Mechanical Engineering, National Taiwan University, Taipei,
Taiwan
* Author for correspondence (e-mail: jtyang{at}ntu.edu.tw)
Accepted 30 July 2008
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Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Key words: fish, pitching stabilization, caudal fin-wave propagation, vortex, stereoscopic-DPIV, parrot cichlid
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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The propagation of a CFP is observable in a parrot cichlid that swims in still water or against a current. When a parrot cichlid initiates a forward sinking, the pectoral fins first induce a head-down pitching moment that tips the head down. Afterwards, the parrot cichlid adopts a stable tilted-down swimming posture (Fig. 1A). In this posture, the transverse amplitude (i.e. lateral displacement) of the dorsally propagating fin wave gradually increases from the ventral to the dorsal edges of the caudal fin.
The hydrostatic pitching equilibrium of a parrot cichlid is normally
neutral because the centers of mass and buoyancy occur at approximately the
same relative longitudinal location of the fish body. For a forward-sinking
parrot cichlid that adopts a tilted-down swimming posture, however, the body
should be inherently unstable in pitching as the center of buoyancy is located
aft of the center of mass. The buoyant force can induce a head-down pitching
moment that destabilizes the fish body but a forward-sinking parrot cichlid
maintains a stable tilted-down swimming posture. The maneuvering of a
forward-sinking parrot cichlid is hence related to a postural control
involving pitching stabilization (Webb,
2002
; Webb, 2004;
Weihs, 1993). To analyze how a
fish achieves this pitching stabilization, one must evaluate the hydrodynamic
forces acting on the fish body, especially the longitudinal aspect
(Weihs, 1993).
There has been little study CFPs. Our objective in this work has been to understand the function of a CFP in a forward-sinking parrot cichlid. We expect that the CFP is related to pitching stability. In this work, we assume that the hydrodynamic forces acting on a fish body are generated by only the caudal and pectoral fins because maneuvers of the other fins are not obvious.
Much important research has focused on the swimming hydrodynamics of
heterocercal and homocercal caudal fins
(Gibb et al., 1999;
Lauder, 2000;
Lauder and Drucker, 2002;
Liao and Lauder, 2000;
Müller et al., 1997;
Nauen and Lauder, 2002a;
Nauen and Lauder, 2002b;
Stamhuis and Videler, 1995;
Wilga and Lauder, 2002;
Wilga and Lauder, 2004;
Wolfgang et al., 1999). Wake
patterns composed of linked vortex rings or a ring-within-a-ring structure
were found. Some homocercal caudal fins are reported to undergo a slight
deformation due to an asymmetric movement of the dorsal and ventral lobes
during maneuvering, so that both thrust and lift are generated
(Gibb et al., 1999;
Lauder, 2000;
Lauder and Drucker, 2002;
Nauen and Lauder, 2002a). The
heterocercal tail in sharks also produces thrust and lift
(Wilga and Lauder, 2002;
Wilga and Lauder, 2004). The
lift generated at the tail can result in a pitching moment about the center of
mass of a fish body (Weihs,
1993).
The pectoral fins of a parrot cichlid invariably move while the fish is
performing a CFP. For instance, as shown in
Fig. 1B, the forward-sinking
parrot cichlid simultaneously abducts its pectoral fin on the right side and
adducts its pectoral fin on the left side. The locomotive function of pectoral
fins has been investigated using kinematic analysis
(Ramamurti et al., 2002;
Walker and Westneat, 1997;
Walker and Westneat, 2002) or
wake-geometry analysis (Drucker and Lauder,
1999; Drucker and Lauder,
2000; Drucker and Lauder,
2001; Drucker and Lauder,
2003). One function of the pectoral fins is to facilitate the
rolling stabilization.
In this work, we evaluated the hydrodynamic forces from the velocity fields
in the fin wake, measured with stereoscopic digital particle-image velocimetry
(commonly abbreviated as stereoscopic-DPIV or SDPIV). Because flow fields
generated by a swimming fish are generally three-dimensional and complicated
(Fish and Lauder, 2006;
Müller and van Leeuwen,
2006; Tytell,
2006; Tytell et al.,
2008), SDPIV is much more accurate than the traditional
two-dimensional DPIV for measuring the flow velocity fields so generated
(Nauen and Lauder, 2002b;
Sakakibara et al., 2004). Like
previous researchers (Drucker and Lauder,
1999; Drucker and Lauder,
2005; Nauen and Lauder,
2002a; Stamhuis and
Nauwelaerts, 2005), we evaluated quantitatively the temporally
averaged locomotive (i.e. hydrodynamic) forces based on a vortex-ring
model.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Stereoscopic digital particle-image velocimetry flow measurement
We used SDPIV to acquire simultaneously, on a light sheet, the in-plane and
the out-of-plane velocities (U, V, W). The two SDPIV cameras (IDT
X-StreamTM Vision 5 high-speed CMOS digital camera, Integrated Design
Tools Inc., Taipei, Taiwan, ROC) were placed in angular type; the angles
between the optical axes (i.e. viewing directions) of each camera and the
light-sheet norm were 30° (Prasad,
2000). Fig. 2 shows
the arrangement of the experimental apparatus for SDPIV and recording images
of the fish in a Plexiglas tank (1 mx1 mx1 m;
heightxwidthxdepth) filled with fresh water. The four side walls
and the bottom of the tank were transparent. The beam from a CW argon-ion
laser (7 W, Spectra-Physics, Stabilite 2017, Fremont, CA, USA) was expanded
into a light sheet (thickness 1.5–2 mm) using cylindrical and spherical
lenses. Nearly neutrally buoyant light-scattering microparticles (hollow glass
spheres: specific gravity, 0.1–1.5; diameter, 8–12 µm; Potters
Industries, Valley Forge, PA, USA) were seeded into the water and illuminated
as tracing particles (Raffel et al.,
2007). The frame rate of the two SDPIV cameras was 250 frames
s–1; the image resolution was 1280x1024 pixels.
Particle image traces were analyzed with software (Insight version 5.0, TSI,
Shoreview, MN, USA) to obtain two-dimensional DPIV velocity vectors. The
procedures of DPIV correlation analysis and vector validation were similar to
those described previously (Drucker and
Lauder, 1999; Nauen and
Lauder, 2002a; Nauen and
Lauder, 2002b). The size of the interrogation window was
16x16 pixels. A Hart correlator (Insight software) was used with 50%
overlap of the interrogation window. For a flow field of 10x10
cm2, approximately 50x50 vectors were computed. The displayed
vectors in the presented vector plots were adequately reduced so as not to
obscure the overlapping color contours. We used backward-mapping second order
polynomials to calibrate the SDPIV images
(Lawson and Wu, 1997;
Lecerf et al., 1999;
Westerweel and Van Oord, 1999;
Willert, 1997). We performed a
calibration-based velocity reconstruction using TPower-SDPIV software written
in Matlab (version 7.0; Mathworks, Natick, MA, USA). The measurement error of
the in-plane velocity components is estimated to be less than 4.2% of the true
values, and of out-of-plane velocity components less than 5.1% of the true
values.
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Measurements of kinematic parameters
To derive kinematic parameters pertaining to the average swimming velocity,
body angle and sinking angle, we filmed and analyzed the motion of a
forward-sinking parrot cichlid. The body angle (
b) is
defined as the angle between the lateral midline of fish body and the
horizontal (see Fig. 1A). The
sinking angle is defined as the angle between the swimming direction and the
horizontal. We filmed the fish motion with the two high-speed cameras used for
SDPIV measurements. One camera filmed the fish motion from a side (i.e.
lateral) view and the other from a top (i.e. dorsal) view. The recorded images
of the side-view camera were analyzed using MotionPro X Studio software (IDT)
to evaluate the kinematic parameters. The top-view camera served only for
simultaneously monitoring the swimming path of the fish.
During the SDPIV experiment on a parasagittal plane, an additional motion camera (Sony HDR-SR1, 60 frames s–1) was placed beneath the water tank (Fig. 2) to film simultaneously the fish motion. The body angle and sinking angle of the forward-sinking parrot cichlid were estimated from the SDPIV image sequences. The backward-mapping polynomials determined in the SDPIV calibration procedure served to correct for the perspective error associated with the swimming path and the body and sinking angles. Otherwise, for the SDPIV experiment on a transverse plane, the camera filmed the fish motion from an approximately side view; the average swimming velocity and body and sinking angles were estimated from these motion-camera images.
Quantitative estimation of forces
Temporally averaged locomotive forces were evaluated quantitatively based
on a vortex-ring model (Drucker and Lauder,
1999; Drucker and Lauder,
2005; Nauen and Lauder,
2002a; Saffman,
1992; Spedding et al.,
1984; Stamhuis and
Nauwelaerts, 2005). The temporally averaged locomotive force (or
central jet force) was calculated using:
 |
is the density of
water (1000 kg m–3 at 20°C), Ar is
the ring area, 
is the absolute mean value of the circulation of the
vortex pair, and Tj is the interval during which the
vortex ring (or jet) is generated. F is essentially an uncorrected jet
force because the calculation is based on the two-dimensional in-plane
velocity fields; to obtain the true jet force Fj, the force
F was further rectified using trigonometric relations between the mean
velocity vector of the central jet and the plane of the light sheet. The mean
velocity vector of the central jet was determined on averaging ten velocity
vectors in the central jet region. To calculate the circulation , we
performed closed-loop integration
(Saffman, 1992) along a path
defined as the zero-vorticity boundary of a vorticity concentration of a
vortex.
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2-value (second-largest eigenvalue of a symmetric tensor
S2+
2) that is a Galilean invariant
(Haller, 2005;
Jeong and Hussain, 1995).
Within a flow field, regions of negative 2 belong to the
vortex core area, and locations of zero 2 correspond to the
boundary of the vortex core.
Measurement of the center of mass
To measure the center of mass of the fish body, we anaesthetized three
parrot cichlids in advance using clove oil. We carefully tied a nylon string
around the unconscious fish and then suspended it
(Fig. 3) for photography. This
procedure was repeated several times with the string in different positions
each time (Fig. 3). After the
measurements, the fish was released into a tank filled with fresh water to
recover.
For a suspended fish, the extension of the upright nylon string definitely passed through the center of mass of the fish; accordingly we located the center of mass by determining the intersection of any two extensions of the upright nylon strings from photographs (Fig. 3C). The center of mass of a parrot cichlid is located approximately, in a lateral view, at a point near the dorsal tip of the pectoral fin base.
Measurement of the center of buoyancy
The same fish used to measure the center of mass were used for the center
of buoyancy measurements. The parrot cichlid is negatively buoyant because an
anaesthetized parrot cichlid sinks when it is dropped into the water. The
weight of the fish hence exceeds the buoyant force. Moreover, an anaesthetized
parrot cichlid in the water has its belly upward, which indicates that the
center of buoyancy is located at a point ventral to the center of mass.
We were unable to determine the exact location of the center of buoyancy,
but found only a straight line on which the center of buoyancy is located. For
a submerged body, the center of buoyancy is well known to be the center of
geometry. As the fish apparently lacks a uniform density, we could not locate
exactly the center of buoyancy by simply determining the center of geometry.
We constructed a simple test section to find the desired straight line
(Fig. 4A); the test section (a
rectangular, transparent water container) was designed to be narrow (width
9 cm), only slightly wider than the lateral body width of the parrot
cichlid, thus preventing the sinking fish body from rolling (rotation about
its longitudinal axis) and yawing (left–right rotation about the
dorsal-ventral axis). The sinking fish was allowed only to pitch (rotation
about the lateral axis of the body with head up or down).
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).
To determine the straight line on which the center of buoyancy was located, we
selected the recorded images for this non-rotating state of the fish body. On
these selected images, the straight line was determined simply on drawing a
vertical line passing through the center of mass of the fish body
(Fig. 4B); this straight line
approximately connected the center of mass and the base of the pelvic fin.
This finding indicates that the centers of mass and of buoyancy occur
approximately at the same relative longitudinal location of the fish body, so
that the fish is normally neutral in a hydrostatic pitching equilibrium
(Weihs, 1993). Although the
exact location of the center of buoyancy is unknown, the center of buoyancy is
definitely located aft of the center of mass in a forward-sinking parrot
cichlid, because the fish adopts a tilted-down swimming posture, implying that
the fish body is unstable in pitching.
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Flow fields and locomotive forces generated by CFP
We acquired in total of 120 SDPIV image sequences in the experiment, from
which 65 sequences free of image occlusion were selected and analyzed to
obtain flow velocity fields (13 sequences for the transverse plane and 52
sequences for the parasagittal plane).
In parrot cichlids a CFP during forward sinking generates propulsive forces involving a negative lift and thrust. The measured near-fin wake revealed that oscillating flow jets consisting of components oriented posteriorly (downstream), laterally and dorsally, designated here as CFP jets, were generated periodically with a caudal fin executing CFP. The dorsally propagating fin wave initially trapped a fluid mass and then accelerated it dorsally. For instance, Fig. 5 shows a representative sequence of four instantaneous flow fields (on a transverse plane) that demonstrate the near-fin wake associated with a CFP; the time was set to t=0 for Fig. 5A. Fig. 5A–D show phases at approximately 29, 50, 60 and 65%, respectively, of a tail beat cycle that was initiated when the dorsal part of the caudal fin was at the last right excursion. The in-plane velocity vectors (Fig. 5A) show two CFP jets (indicated with bold white arrows) that occur near the dorsal and ventral parts, respectively, of the caudal fin. The dorsal (upper) CFP jet was formed earlier than the ventral (lower) one. The dorsal CFP jet was approaching a state of being expelled. The fluid mass of the ventral CFP jet was initially trapped by the fin into a concave zone of low pressure that was created by the lateral movement of the ventral part of the caudal fin, according to the principle of conservation of mass.
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In particular, vortex 2 did not subsequently continue to grow. The in-plane velocity vectors in Fig. 5B–D show that vortex 2 in Fig. 5A was destroyed and gradually weakened by a newly formed vortex 3 (in Fig. 5C). The formation of vortex 3 results from the reinforcement of the ventral CFP jet.
At t=0.16 s approximately (Fig. 5B), the dorsal CFP jet began to be expelled. The in-plane velocity magnitudes were smaller on average at the ventral CFP jet than at the dorsal CFP jet (Fig. 5A–D). The ventral CFP jet was more dorsally oriented in Fig. 5D than in Fig. 5A, indicating that the caudal fin had imparted dorsally oriented momentum to the jet and correspondingly obtained a ventrally oriented reaction force. The ventral CFP jet eventually evolved into a dorsal CFP jet and became expelled, with a lateral component of orientation opposed to that of the dorsal CFP jet in Fig. 5.
The flow fields shown in Fig. 5 were not on an exactly transverse plane as an angle of approximately 28° existed between the swimming direction of the fish and the y–z plane. This condition, however, did not affect the identification of the flow topology of the near-fin wake on a transverse plane. Our results indicate that alterations of the flow pattern of the near-fin wake were correlated mainly with the phases of the caudal-fin beat cycle.
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2-value and vorticity corresponding
to the velocity field in Fig.
5C are shown in Fig.
6. The locations designated Vx1,
Vx2, Vx3 and Vx4
(Fig. 6A) are, respectively,
vortex cores of vortices 1, 2, 3 and 4
(2=–52.0, –37.9, –51.6, –2.29,
respectively). The 2-value contour confirmed the
identification and location of vortices 1, 2, 3 and 4. The
vorticity contour (Fig. 6B)
confirmed the occurrence of CFP jets that were generated by the caudal fin
because the intermediate region between a clockwise vortex (with negative
vorticity) and a counterclockwise vortex (with positive vorticity) are
recognizable as a jet flow. The two-dimensional in-plane flow pattern
comprising a counter-rotating vortex pair and its central jet is essentially a
vortex-ring-like structure three-dimensionally. Fig. 7 shows the near-fin wake measured on a parasagittal (x–y) plane of the light sheet that was approximately parallel to, and overlapping, the middle plane of the fish body (see the schematic drawing at the top of Fig. 7). Fig. 7A and B are two instantaneous velocity fields within a stroke cycle. Fig. 7C and D are respectively the corresponding vorticity contours for Fig. 7A and B. For Fig. 7A and C, the dorsal part of the caudal fin was beating laterally toward the positive z-direction, while the ventral part of the caudal fin beat was toward the negative z-direction. The in-plane velocity vectors (Fig. 7A,C) reveal the dorsal and ventral CFP jets generated. The dorsal CFP jet was formed earlier than the ventral one. The dorsal CFP jet having posterodorsally oriented in-plane velocity components and positive W (out-of-plane) velocity components was being expelled near the dorsal part of the caudal fin (Fig. 7A,C). The ventral CFP jet, having anterodorsally oriented in-plane velocity components and negative W velocity components, was approximately in a stage of initial formation.
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In Fig. 7B,D, the dorsal part of the caudal fin was beating laterally toward the negative z-direction, while the ventral part of the caudal fin toward the positive z-direction. The ventral CFP jet observed in Fig. 7A,C was being expelled near the dorsal part of the caudal fin (Fig. 7B,D). The CFP jet (between vortices P2 and P3) comprised components oriented dorsally, posteriorly and laterally.
Table 1 presents the CFP jet
forces calculated with flow data measured from the parasagittal
(x–y) plane of the light sheet (cf. the schematic drawing at
the top of Fig. 7). The
circulation used to calculate the jet force was taken as the average
of absolute for vortices P1 and
P2 or vortex P3 as the CFP jet was
expelled. Both the mean vertical force (Fjy) and lateral
force (Fjz) were approximately four times the mean
horizontal force (Fjx)
(Table 1). The lateral forces
would cancel each other because the caudal fin beat periodically and the CFP
jets were oscillating laterally. In sum, the CFP in a forward-sinking parrot
cichlid produces mainly a negative lift and a small thrust. As the caudal fin
is located posterior to the center of mass of the fish body, the force
generated via CFP can result in a head-up pitching moment that lifts
up the fish head.
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Flow fields and locomotive forces generated by pectoral fins
Within a complete stroke cycle, motions of the maneuvering pectoral fins of
a forward-sinking parrot cichlid executing a CFP involved sequential fin
movements of (1) anteroventrally oriented fin abduction (i.e. the
down-stroke), (2) rotating and cupping the fin (i.e. stroke reversal) and (3)
posterodorsally oriented fin adduction (i.e. the up-stroke). The results of
the flow measurement reveal that the down-stroke of the pectoral fin created a
fluid jet (designated a down-stroke jet) comprising components oriented
posteriorly, laterally and ventrally (see
Fig. 8). A head-down pitching
moment was generated by the pectoral fins during the down-stroke. The
up-stroke of the pectoral fin also created a fluid jet (designated an
up-stroke jet) comprising components oriented posteriorly, laterally and
ventrally.
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Fig. 8 shows the wake of the pectoral fin measured on a parasagittal (x–y) plane of the light sheet approximately parallel to the fish body and intersecting the pectoral fin (see the schematic drawing at the top of Fig. 8). Fig. 8A–C are instantaneous velocity fields of the wake for down-stroke, stroke reversal and up-stroke, respectively, within a stroke cycle. Fig. 8D–F are the corresponding vorticity contours for Fig. 8A–C, respectively. Fig. 8A demonstrates that a three-dimensional down-stroke jet was created at the end of the down-stroke. A counter-rotating vortex pair (K1 and K2) adjacent to the down-stroke jet was observable from the in-plane velocity vectors and the vorticity contour (Fig. 8D). The in-plane component of the down-stroke jet indicates that a head-down pitching moment was induced because the pectoral fin was located posterior to the center of mass of the parrot cichlid during the down-stroke.
During stroke reversal, both the in-plane and out-of-plane velocity fields indicate that a new fluid jet was ready to form (Fig. 8B). The formation of a new vortex K3 was initiated (Fig. 8E). The in-plane velocity vectors between vortices K1, K2 and K3 formed a curved jet (represented by the bold curved black arrow in Fig. 8E).
Fig. 8C shows the upstroke jet formed at the end of the up-stroke. The up-stroke jet was more posteriorly oriented than the down-stroke jet. The in-plane component of the up-stroke jet passed approximately through the center of mass of the fish body; the pectoral fin was thus considered not to generate a pitching moment during an up-stroke. Vortices K3 and K2 with counter-rotating directions were the vortex pair adjacent to the up-stroke jet.
Table 2 presents the
down-stroke and up-stroke jet forces calculated using flow data measured from
the parasagittal (x–y) plane of the light sheet (cf.
the schematic drawing at the top of Fig.
8). The circulation used in the jet force calculation was
taken as an average of absolute for vortices K1
and K2 at the end of the down-stroke, and the average of
absolute for vortices K2 and
K3 at the end of the up-stroke. The total jet forces of
the down-stroke were approximately twice as large as that of the up-stroke.
Both down-stroke and up-stroke produced large lateral forces
(Fjz). The up-stroke produced a considerable thrust and a
small negative lift. The thrust produced by an up-stroke was larger than that
produced by a down-stroke.
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DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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;
Webb, 2004;
Weihs, 1993). Our experimental
results indicate that the caudal fin executing a CFP can generate CFP forces
(see the FCFP in Fig.
10) that induce a head-up pitching moment. As the swimming posture
of the fish is stable, one can reasonably infer that those head-down pitching
moments that destabilize the fish body are balanced approximately by the
head-up pitching moment induced from a CFP (see
Fig. 10). The behavior of a
CFP functionally facilitates the pitching stabilization in a forward-sinking
parrot cichlid that adopts a tilted-down swimming posture.
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For a parrot cichlid, the magnitude of the buoyant force is typically larger than that of the hydrodynamic forces generated by a fish's caudal and pectoral fins (the buoyant force of a fish is estimated roughly as the weight of water displaced by a submerged fish, in a graduated measuring cup). The moment arm associated with the buoyant force is hence much shorter than those associated with the CFP force or the down-stroke force of pectoral fins. For instance, the buoyant force of a parrot cichlid used in this work is approximately 1.6 N, but the hydrodynamic forces generated by a fish's fins are in a range approximately 1–10 mN (cf. Tables 1 and 2). To balance all the induced pitching moment, the magnitude of the head-down pitching moment induced by the buoyant force should be comparable with that of the hydrodynamic forces generated by a fish's fins. As the center of buoyancy is known to be located on a straight line connecting the center of mass and the base of the pelvic fin (see Fig. 4), one can further infer that the center of buoyancy is located at a point (on the straight line) near the center of mass.
The functional characteristics of CFP behavior might provide useful insight into the locomotive function of highly flexible or deformable appendages of aquatic animals, and would be also beneficial for the design of autonomous underwater vehicles with regard to control of pitching stability.
Structural characteristics of the CFP wake
Fig. 11 shows sequential
two-dimensional schematic drawings that summarize the formation and evolution
of CFP jets and vortices observable in the near-fin wake of a forward-sinking
fish executing CFP; these CFP jets are three-dimensional (cf.
Fig. 12). The flow patterns of
the near-fin wake shown in Fig.
11A–D correspond to those observed at approximately the 1/3,
1/2, 4/5 and end phases, respectively, of a single CFP beat cycle. Alterations
of the flow pattern of the near-fin wake are correlated mainly with the phases
of the beat cycle of the caudal fin.
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When observed from behind the fish (Fig. 11), the formation and evolution of a developing CFP jet can be divided approximately into three stages. A CFP jet is induced (or trapped) initially by a zone of low pressure created by the fin movement near the ventral part of the caudal fin (see Jet 2 in Fig. 11A). In this stage, the developing CFP jet orients approximately toward the direction of movement of the ventral part of the caudal fin. A CFP jet (see Jet 1 in Fig. 11A) formed earlier is expelled near the dorsal part of the caudal fin. Afterwards, the propagating fin wave gradually reorients and accelerates the developing CFP jet (Jet 2 in Fig. 11B,C). The caudal fin continuously imparts momentum to the fluid while the fin wave travels dorsally. In the second stage, the CFP jet formed earlier is almost completely expelled (Jet 1 in Fig. 11B), and the formation of a new CFP jet is initiated (Jet 3 in Fig. 11C).
In the third stage, the developing CFP jet eventually becomes a strong flow jet and is completely expelled near the dorsal part of the caudal fin (Jet 2 in Fig. 11D). In brief, a CFP jet, for example Jet 2 in Fig. 11, is initially induced (or trapped) by the ventral part of the caudal fin, subsequently strengthened and reoriented by the propagating fin wave, and eventually expelled near the dorsal part of the caudal fin.
We propose three-dimensional structures of the near-fin and far-fin wake of
the CFP (see Fig. 12A,B). The
three-dimensional near-fin wake presented in
Fig. 12A structurally
corresponds to the two-dimensional case shown in
Fig. 11B. The far-fin wake
(Fig. 12B) comprising linked
vortex rings and oscillating central CFP jets strongly resembles a `reverse
von Karman vortex street' that is observable in the wake of carangiform
swimmers or biomimetic foils (Lugt,
1995; Nauen and Lauder,
2002a; Nauen and Lauder,
2002b; Triantafyllou et al.,
2000; Triantafyllou et al.,
2004). The two-dimensional sketch at the upper right of
Fig. 12B illustrates the
vortex filaments of the far-fin wake observed on a parasagittal plane.
We observed that an expelled CFP jet had a slightly divergent configuration. The dorsally propagating fin wave was initiated by the lateral movement of the ventral part of the caudal fin, which could first trap the fluid volume of a CFP jet and then expel it. The volume of the ventral CFP jet observed in Fig. 7A is divisible into approximately two parts: one, the right part, was nearer the ventral region of the caudal fin (or vortex P3), whereas the other, the left part, was more distant from the ventral region of the caudal fin and nearer vortex P2. The right part was directly accelerated by the fin displacement, whereas the left part was indirectly accelerated due to the fluid viscosity. While the ventral CFP jet was developing, the left part was expelled (or ejected) earlier than the right part, because the left part was farther from the caudal fin and hence left the influence of the caudal fin earlier. The left part would convect downstream according to its fluid inertia. The right part having not been expelled would be further propelled dorsally by the dorsally propagating fin wave, and become expelled eventually near the dorsal edge of the caudal fin. The in-plane velocity vectors (Fig. 7B) indicate the right part is more dorsally oriented than the left part, as the right part is dorsally propelled by the caudal fin for a greater duration. The CFP jet consequently showed a slightly divergent configuration as it was completely expelled (Fig. 7B).
A slightly divergent CFP jet is considered to be structurally similar to a
split jet generated by the shark tail possessing an asymmetric shape
(Wilga and Lauder, 2004). A
split jet corresponds to a `ring-within-a-ring' structure in the wake and is
proposed to increase the vertical maneuverability of a shark
(Wilga and Lauder, 2004). We
infer that the slightly divergent CFP jet may also have a similar function of
increasing the maneuverability. The CFP jet force may be adjustable by tuning
separately the orientation of the two divergent parts (or components) of the
jet.
Based on qualitative estimation, we found the volume of a CFP jet was
generally increased when it was expelled. For the `in-plane' aspect of flow
fields (e.g. Fig. 7A,B), the
volume variation of a CFP jet can be qualitatively estimated from an
examination of the variation of the distance between the cores of vortex pairs
of the CFP jet recognized by the 2-value
(Jeong and Hussain, 1995). As
this distance invariably increased, in the in-plane aspect the volume of a CFP
jet increased when it was expelled. Otherwise, for the `out-of-plane' aspect,
the variation in volume of a CFP jet can be qualitatively estimated from an
examination of the variation of the `in-plane' area subject to negative
W velocity (i.e. the out-of-plane velocity). As this area invariably
increased, in the out-of-plane aspect the volume of a CFP jet increased when
it was expelled. We consider that the gradually increasing transverse
amplitudes of the caudal fin wave gradually enlarge the fluid volume of a
developing CFP jet due to gradually enlarged extent of interaction between fin
and fluid.
In this work we also noticed that the motion of a CFP probably enabled the
fish to recycle the energy imparted from a self-shed vortex. This is because
the vortex or vortex loop pertaining to the dorsal CFP jet is spatially near,
or even overlaps, the ventral CFP jet that has approximately the same velocity
direction as the local vortex motion. Flow fields measured from both the
transverse (Fig. 5) and
parasagittal (Fig. 7) planes
showed that the vortex loop pertaining to the dorsal CFP jet partly
overlapped, or was near, the ventral CFP jet. The vortex loop of the dorsal
CFP jet in Fig. 5 involved
out-of-plane flows (with negative W velocity) that overlapped the
ventral CFP jet; the counterclockwise circulatory flow of vortex 2 in
Fig. 5 also partly overlapped
the ventral CFP jet in the in-plane aspect. Vortex P2 in
Fig. 7C was structurally shared
by the dorsal and ventral CFP jets. The schematic drawing in
Fig. 11A illustrates that a
vortex b is structurally shared by the dorsal and ventral jets (i.e. the Jet 1
and Jet 2). The circulation of a vortex or vortex loop theoretically indicates
a flow angular momentum that can drive (because of fluid viscosity) nearby
fluid mass flowing in the same direction of rotation of the vortex or vortex
loop (Munson et al., 1998).
The shed vortex (or vortex loop) pertaining to a dorsal CFP jet evidently
assists the trapping of the fluid mass of the subsequently developing ventral
CFP jet (Fig. 5). Since the
energy (or momentum) involved in the circulation of the shed vortex was
imparted by the fish, the energy was thus probably recycled on trapping the
fluid mass, which is functionally similar to the vortex-based extraction of
energy by aquatic animals that extracts energy from environmental vortices
(Allen and Smits, 2001;
Dabiri, 2007;
Liao et al., 2003).
Functions of the pectoral fins
During an up-stroke, the pectoral fins generate considerable thrust and a
small negative lift (cf. Table
2). Because the pectoral fins generate no pitching moment during
an up-stroke, it is thought that the principal function of an up-stroke of a
pectoral fin is to provide thrust, but the pectoral fin generates a head-down
pitching moment during a down-stroke. This head-down pitching moment is
detrimental to the pitching stability of a fish body as it cannot counteract
the head-down pitching moment induced at the center of buoyancy. In summary,
the pectoral fin of the parrot cichlid destabilizes the fish body in the
aspect of pitching.
The lateral forces generated by pectoral fins are large (cf.
Table 2), and are considered
capable of stabilizing the fish body and preventing it from rolling, as
reported previously (Drucker and Lauder,
1999). The lateral forces would approximately cancel, considering
contributions of pectoral fins at both sides.
LIST OF ABBREVIATIONS
22
b
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Acknowledgments |
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References |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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