Fluid dynamics of flapping aquatic flight in the bird wrasse: three-dimensional unsteady computations with fin deformation
1 Laboratory for Computational Physics and Fluid Dynamics, Naval Research
Laboratory, Washington, DC 20375-5344, USA
2 Computational Sciences and Informatics Department, George Mason
University, Fairfax, VA 22030, USA
3 Department of Biological Sciences, University of Southern Maine, Portland,
ME 04103, USA
4 Department of Zoology, Field Museum of Natural History, Chicago, IL
60605-2496, USA
Accepted 9 July 2002
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Summary |
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Key words: bird wrasse, Gomphosus varius, fin kinematics, flapping aquatic flight, unsteady flow, unstructured mesh, deforming fin
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Introduction |
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In their earlier work on the mechanics of flapping fin propulsion in
fishes, Walker and Westneat
(1997) inferred fin
hydrodynamics by comparing detailed fin kinematics with measures of
center-of-mass accelerations throughout the fin stroke cycle. They used the
center of mass accelerations in place of direct measurements of the
instantaneous force balance, because the latter cannot be measured on a freely
swimming animal. The vector of instantaneous center-of-mass acceleration
differs from the vector of net forces at the center of mass by a constant,
hence the pattern of thrust and lift occurring throughout the stroke will be
the same, regardless of whether this is directly measured with a force
transducer or estimated by center-of-mass kinematics. More recently, the
dynamics of pectoral fin propulsion in the bluegill Lepomis
macrochiris and surfperch Embiotoca jacksoni was investigated
using digital particle image velocimetry (DPIV) of the wake (Drucker and
Lauder, 1999
,
2000
). These wake studies have
proved useful in exploring the fluid dynamic events occurring at the fin.
In this study, we seek to complement these experimental studies of fin kinematics, center-of-mass dynamics and wake visualization by computing the unsteady flow about G. varius with pectoral fin oscillation and deformation prescribed from the experimental kinematics. We continue our earlier computational focus on oscillating control surface flows for non-undulating bodies, using G. varius. The primary objectives in this work are to (i) investigate the fluid dynamics underlying the generation of forces during pectoral fin oscillation, (ii) compare the hydrodynamic utility of steady, quasi-steady and unsteady hydrodynamic models of fin propulsion, and (iii) to compare the fluid dynamics of a flapping appendage during forward motion with a flapping appendage during hovering.
For this last objective, we note that the results are relevant not only to
fish propulsion but also to the locomotion of any animal moving with
oscillating appendages in a similar fluid-dynamic environment. For example,
G. varius at the large end of the size range investigated by Walker
and Westneat (1997,
2002a
,b
)
has approximately the same mass and flaps with approximately the same reduced
frequency and Reynolds number as M. sexta
(Willmott et al., 1997
). Both
G. varius and M. sexta wings are stiff along their span.
G. varius flaps its wings along a steeper stroke plane than M.
sexta, although at the high end of the latter's flight speed these values
converge. Finally, while the wing of M. sexta has a greater aspect
ratio than that of G. varius, the radial moments of area are the same
(see corrected values for G. varius in
Walker and Westneat,
2002b
).
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Materials and methods |
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Unstructured mesh generation and adaptive re-meshing
In order to carry out computations of the flow about oscillating and
deforming geometries, which may be quite complex, several pieces of grid
technology are needed. First, one must be able to rapidly generate a surface
triangulation. If many complex surfaces are intersecting and they are discrete
components, as is the case for multiple fins on a fish body, it is essential
to be able to construct the total surface mesh, including the intersection
loci, automatically. One then needs to describe the mesh motion on the moving
surface, couple the moving surface mesh to the volume grid in a smoothly
varying manner, and describe the dynamic remeshing of the volume grid in
proximity to the moving surface as the surface moves and deforms. In
deformations, the surface motion may be severe, leading to distorted elements
in the absence of remeshing, which in turn lead to poor numerical results. If
the bodies in the flow field undergo arbitrary movement, a fixed mesh
structure will lead to badly distorted elements. This means that at least a
partial regeneration of the computational domain is required. On the other
hand, if the bodies move through the flow field, the positions of relevant
flow features will change. Therefore, in most of the computational domain a
new mesh distribution will be required.
One approach to solving these problems is to add several layers around the moving bodies that move rigidly with the body. As the elements (or edges) move, their geometric parameters (shapefunction derivatives, jacobians, etc.) need to be recomputed at every timestep. If the whole mesh is assumed to be in motion, then these geometric parameters need to be recomputed globally. In order to reduce the number of global remeshings and hence save computational time when using this approach, only a small number of elements surrounding the bodies are actually moved. The remainder of the field is then treated in the usual Eulerian frame of reference, avoiding the need to recompute geometric parameters. We refer the reader to earlier paper (Löhner, 1998) that discuss the mathematics and numerics of the unstructured grid generation and adaptive remeshing codes used in this work.
Fish
The G. varius Lacepède 1801 individual was acquired from a
tropical fish wholesaler and maintained in a 2281 aquarium within a 23001
recirculating marine system until it was euthanised with an overdose of MS-222
(Finquel brand, Aldrich Chemical Co.) and frozen at -20°C.
Three-dimensional wrasse body and pectoral fin description
To obtain the 3-D surface coordinates of a bird wrasse, an individual of
standard length L=21 cm was frozen and sliced into nine transverse
sections. Section outlines were digitized using a modification of the public
domain NIH Image program (developed at the US National Institutes of Health
and available on the Internet at
http://rsb.info.nih.gov/nih-image/)
for the Apple Macintosh (the modification is available upon request from J. A.
Walker). The outline coordinates were used to generate a smooth surface using
standard cubic spline methods. While the digitized individual is
representative of the geometry of the subjects from the experiment
(Walker and Westneat, 1997),
it was not one of the experimental subjects. The exact geometry and the
corresponding surface mesh of the bird wrasse for which the computations were
done are shown in Fig. 1.
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Pectoral fin kinematics data
Pectoral fin surface coordinates were estimated from the experimental data.
In the original experiment of Walker and Westneat
(1997), five aluminum markers
were attached to one of the pectoral fins: two on the leading edge, two on the
trailing edge and one on the fin tip. Fin motion was filmed using S-VHS
videotape at 60 Hz. The five fin markers and the dorsal base of the pectoral
fin were digitized from both lateral and dorsal views. The 3-D coordinates of
the markers throughout the cycles were obtained from the marker positions in
the two views. For the present analysis, the motion of the three distal-edge
markers was smoothed with a quintic spline function
(Walker, 1998
). In order to
remove the kink in the distal (tip) edge of the fin that necessarily resulted
from having only three digitized points, a smooth curve was fitted to the
distal edge by increasing the number of points to 14 (the number of fin rays
in the fin), using linear interpolation and smoothing the distal edge with a
quintic spline function (Walker,
1998
). 3-D surface coordinates were obtained by linear
interpolation between the digitized pectoral fin base of the representative
individual, scaled to the size of an experimental individual, and the distal
edge of an experimental individual.
Walker and Westneat (1997)
observed that the fins flapped synchronously during rectilinear motion at all
test speeds. They also noted that G. varius flapped its pectoral fins
up and down with a small anterior movement during abduction and a small
posterior movement during adduction. Flapping frequency was seen to increase
linearly with speed. The mean flapping frequency was 2.9 Hz at 22 cm
s-1 and 4.2 Hz at 50 cm s-1. In the computations carried
out here, we used a pectoral finoscillation frequency of 3.3 Hz, corresponding
to a swimming speed of approximately 45 cm s-1 (2 L
s-1). The digitized positions of the five markers throughout the
oscillation cycle served to specify the kinematics of the pectoral fin for the
present computations. Photographs showing dorsal and lateral views of fin
geometry at two extended positions during the cycle are shown in
Fig. 2.
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Results and Discussion |
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Quasi-steady computations
A set of quasi-steady computations were carried out, as an intermediate
step, to determine the differences between steady state computations for the
fin at fixed positions and angles of attack and computations also
incorporating fin kinematics data. The positions of the pectoral fin at
selected times during both abduction (downstroke) and adduction (upstroke),
shown in Fig. 4, were chosen
for the quasi-steady computations. To simulate the quasi-steady state solution
at any instant of time, the velocity of the fin was obtained from the
experimental kinematics data at that instant. This velocity was then used as
the mesh velocity at the fin surface, without actually moving the fin surface.
The computed flow solution is thus the steady state flow with the high
velocity motion of the fin superimposed on it. (Note that this is not the
usual approach taken for what is termed `quasi-steady' in the literature.
Usually the flow is merely computed about the geometry at successive angles of
inflow, using the same steady inflow velocity for each angle of attack, thus
neglecting the induced velocity due to the fin kinematics.) Forces on the
pectoral fin and the fish body were computed, for the quasi-steady simulation,
for each orientation of the fin, by integrating the surface pressures.
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Comparison of the steady flow forces and those from the quasi-steady flow computation (Fig. 5) shows a considerable difference. Fig. 5A shows the coefficients of the thrust and lift obtained from the steady state computation at several time instants throughout the stroke cycle. The steady state flow was computed with the fin fixed at the corresponding position with the mesh velocity, wfin=0. The computations resulted in a net drag (as opposed to thrust from the quasi-steady computations) for all the orientations considered throughout the stroke cycle. The computed lift is negative for fin orientations up to t=0.2 s and is positive thereafter.
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We next investigated whether either of these computations was in agreement
with the experimental data. The trend of the quasi-steady computed forces,
Fig. 5B, reproduces the trend
of the experiments. For example, the center-of-mass acceleration data
(fig. 5 of
Walker and Westneat, 1997)
indicates that there is a large upward vertical force during the downstroke, a
large negative vertical force during the upstroke, and a large positive thrust
during the upstroke. In addition, it was found in the experiments that the
maximum upward vertical force during abduction occurred between 30-40% of the
abduction phase, the maximum downward vertical force during adduction occurred
at approximately 40% of the adduction and the maximum thrust also occurred at
40% of the adduction phase. All of these maxima are observed in
Fig. 5B to occur at about the
same times in the computational cycle, as was observed in the swimming
experiments. This agreement between the experimental and computed times and
signs of force maxima and minima is to be expected if accurate experimental
kinematics are incorporated correctly into the quasi-steady computation.
Assuming the mass of the fish to be 100 g, the peak vertical acceleration from
the computed forces is 7.5 cm s-2 during the abduction phase and 15
cm s-2 during the adduction phase. The experimental peak values are
in the range 70-100 cm s-2. From the quasi-steady computed forces,
the peak foreaft acceleration is 10.5 cm s-2, compared to
the experimental range of 40-80 cm s-2. Hence, the magnitudes of
the quasi-steady computed forces are not correct. Thus, true quasi-steady
computations are useful but, because 3-D inertial forces are neglected, not
sufficient. Unsteady computations must be performed.
Unsteady computations
Unsteady computations were also carried out using the prescribed fin
kinematics. A new mesh-movement capability, to accommodate the deforming fin
surface, was developed and added to the mesh-movement algorithm used in
earlier flapping-fin computations (Ramamurti and Sandberg,
2001,
2002
; Sandberg and Ramamurti,
2001). The motion of the fin surface was first prescribed at a finite set of
control points. The Cartesian coordinates on the fin surface were then
transformed to a parametric space. The coordinates of the surface points were
maintained constant in the parametric space throughout the computation, while
the Cartesian coordinates were computed according to the prescribed motion of
the control points.
Unsteady simulations were carried out with the bird wrasse swimming at 45 cm s-1. The stroke amplitude is approximately 2.14 rad and the frequency of fin oscillation is 3.3 Hz, resulting in a mean tip speed of approximately 50 cm s-1. The computation was carried out for more than four cycles of fin oscillation using a computational mesh consisting of approximately 150x103 points and 840x103 tetrahedral elements.
At the beginning of the downstroke, the fin is quite close to the body and it is difficult to clearly visualize the fin flow field, hence these time steps have not been included. The simulation of the pectoral fin motion was actually begun slightly after the start of the downstroke, at t=0.05 s, rather than at t=0 s, to avoid the difficulties associated with contact of the fin and the body surfaces. This is considered to have almost no influence on the magnitude of the forces, since the experimentally observed foreaft accelerations are in the range 0-20 cm s-2. A NavierStokes computation to estimate the pressure and viscous drag on the body yielded a total drag of 1.6 kg. The aft acceleration for a 100 g fish is 16 cm s-2, suggesting that the force production due to the fin at this instant is negligible.
The time-varying 3-D lift and thrust (Fig. 6) were computed by integrating the surface pressure over the wrasse body and fin at each time step throughout the simulation. It was noticed that the experimentally specified kinematics at the point where the fin is closest to the fish body were not continuous. This is because the downstroke is steeper than the upstroke, and occurs at the end of the upstroke. This discontinuity, however, does not alter the essential aspects of force production and hence constitutes only a minor perturbation to the results. The computed lift and thrust from the unsteady simulations (Fig. 6) are much larger than those from the quasi-steady simulation (Fig. 5). The peak thrust from the unsteady computations is 0.045 N, compared to 0.01 N in the quasi-steady computations. The unsteady lift varies between -0.06 N and 0.08 N, while those for the quasi-steady computation range between -0.015 N and 0.0075 N.
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The velocity vectors on the entire wrasse and its pectoral fin are shown in Fig. 7A. The body flow at this instant is typical of that observed throughout the stroke cycle. The flow over most of the wrasse body is observed to be uniform throughout the stroke cycle, with a recirculating flow region at the junction of the pectoral fin and the body. The highest velocities are observed on the pectoral fin, above the root of the fin, and on the dorsal side just anterior to the caudal fin.
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A closer view of the pectoral fin flow is given in Fig. 7BF at critical times during the cycle. Fig. 7B shows the flow velocity vectors at t=0.963 s (32% of the downstroke), when the thrust reaches a maximum. In addition to the junction vortex, we see the wake of the previous upstroke on the body downstream of the fin. Fig. 7C shows the velocity vectors on the pectoral fin at t=1.065 s, which occurs at approximately 84% through the downstroke. This is the time of minimum total force production on the downstroke. A clockwise flow is seen at the root above the leading edge and a large recirculation region is present between the fin and the body. In addition to this separated flow region, a small recirculating region can be seen at the junction of the trailing edge of the fin and the body. Fig. 7D shows the velocity vectors occurring at t=1. 1021 s, which corresponds approximately to the beginning (8%) of the upstroke, and the magnitude of the velocity on the upper surface has increased. We observe that, despite the counterclockwise flow on the surface behind the leading edge near the fin root, the clockwise flow on the wrasse body above the root remains. Fig. 7D shows the velocity vectors at t=1.140 s, corresponding to 43% of the upstroke; the recirculation region from the leading edge of the fin is elongated along the body and the region on the surface of the fin is reduced. Also, the vortex near the body-trailing edge junction has formed again. At the instant when the thrust is maximum during the upstroke, t=1.167 s (68% of the upstroke), the interaction of the wake from the upstroke and the body can be seen in the low velocity regions on the body.
In order to provide additional information on the flow about the fin during times of peak force production, we also examined the surface pressure contours. Fig. 8 shows the surface pressure contours, in N m-2, on the pectoral fin at three instants when the peak in the thrust occurs. At t=0.963 s (32% of the downstroke) (Fig. 8A,B), maximum and minimum pressures occur below and above the leading edge of the fin, respectively, producing maximum thrust (in the -x direction) and lift (in the +y direction). At t=1.065 s (84% of the downstroke) (Fig. 8C,D), maximum and minimum pressures occur above and below the leading edge, respectively, in the outer half-span of the fin, producing minimum thrust. At t=1.167 s (68% of the upstroke) (Fig. 8E,F), a high pressure region extends for more than half of the fin on the dorsal side, while the pressure on the ventral side is almost uniform with a region of minimum pressure near the outer leading edge, producing a maximum thrust.
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The wake of the pectoral fin in the z=1.5 cm plane is shown at
critical instants during the oscillation in
Fig. 9. The swimming velocity
of the fish (45 cm s-1) is subtracted from the x component
of the velocity to reveal the vortical structures. These are qualitatively
similar to the patterns observed by Drucker and Lauder
(1999,
2000
) in the wake of a
surfperch. A quantitative analysis of the wake vortex structure can be
performed after a NavierStokes computation of the flow and will be
presented in a later work. At the beginning of the downstroke, two
counter-rotating vortices are observed at t=0.906 s,
Fig. 9B. These vortices are
shed from the distal edge on the previous upstroke. A large leading edge
vortex spanning the entire chord is observed on the suction side in
Fig. 9C at t=1.065 s,
(84% of the downstroke). The thrust reaches a minimum at this instant.
Fig. 9D shows a vortex being
shed from the trailing edge at t=1.14 s (43% of the upstroke). During
the middle of the stroke, the distal edge acts like the trailing edge, hence,
we term this a `trailing edge' vortex. The shedding of this trailing edge
vortex into the wake leads to a momentary increase in lift
(Fig. 6A). At t=1.167
s (68% of the upstroke), the trailing edge vortex is convected downstream.
Also, the chord at this instant is aligned so that the high pressure produced
by the leading edge vortex on the dorsal side of the fin produces a maximum
thrust. Prior to stroke reversal, at t=1.197 s, two vortices are shed
from the distal edge.
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As an additional diagnostic for the flow at the leading edge we plotted the
instantaneous particle traces in the middle of the downstroke
(Fig. 10B) and just after the
stroke reversal (Fig. 10C).
Particles are released from a rake of rectangular grid of points 0.75 cm away
from the leading edge of the fin and parallel to it, as shown in
Fig. 10A. We observed a vortex
on the ventral side of the fin just after stroke reversal
(Fig. 10C). There was no
indication of a strong spanwise flow, which would exist if a leading-edge
spiral vortex was present. Our flapping wing computations for Drosophila
melanogaster (Ramamurti and Sandberg,
2002) also showed no evidence of a spiral leading-edge spanwise
vortex. This suggests that the dynamics of force generation in the pectoral
fin of the swimming wrasse is different from the fluid dynamics of force
production in the hovering hawkmoth. A leading-edge spiral vortex with
spanwise flow was seen by Ellington et al.
(1996
) in their wind-tunnel
experiments on tethered hawkmoths. The spanwise flow in the vortex core has
been proposed as the mechanism for stabilization of the leading edge vortex.
In our case, as can be seen from the surface velocity vectors and particle
traces, we are clearly more dominated by the axial flow than is the case for
the hovering hawkmoth.
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We also computed the foreaft and dorsoventral accelerations
of the fish, assuming a mass of 100 g, from the thrust and lift
(Fig. 11), and compared them
with the experimentally derived foreaft acceleration results for the 47
cm s-1 data from fig.
5A of Walker and Westneat
(1997). We found the
computational results to be in good agreement with the data from the swimming
experiments. The principal observations from the comparison between the
computed results and those from the experiments are presented below.
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Axial forces and accelerations
An aft acceleration is observed experimentally throughout the downstroke
for the 47 cm s-1 data set. The maximum value of the aft
acceleration, which ranges between 20-45 cm s-1, occurs at
approximately one quarter of the downstroke, and again just after the
three-quarters region of the downstroke. The aft acceleration decreases to
approximately zero near midstroke, and again at the end of the downstroke. The
acceleration data for the other swimming speeds all show a small acceleration,
which is aft-directed at first, then changes sign but has a small magnitude,
and then becomes aft-directed for the remainder of the downstroke (see
fig. 5A of
Walker and Westneat,
1997).
The computed axial acceleration starts near zero (Fig. 11A) as the downstroke begins and increases to a maximum at about 30% of the downstroke. It then decreases, going through zero acceleration to an aft-directed acceleration for the remainder of the downstroke. The maximum aft-directed acceleration occurs at about 90% through the downstroke, in excellent agreement with the data. The small forward-directed acceleration peak that we observe from the computations is not present in the data. Adding the viscous body and fin drag to our computed drag values reduces all the positive accelerations and increases the aft-directed accelerations. We made a conservative estimate of the body and fin viscous drag throughout the swimming cycle by carrying out a NavierStokes drag computation for the fully extended fin case. The Reynolds number for this simulation is set to be approximately 16,000, based on the pectoral fin length and a fish swimming speed of 45 cm s-1. The fully extended pectoral fin configuration corresponds to an instant during the middle of the downstroke. This computation yields a body viscous drag of 1.05x103 g and a fin viscous drag of 2.20x103 g, which corresponds to an aft-directed axial acceleration for the 100 g fish of 32.5 cm s-2, which exceeds the magnitude of the positive peak. This value cannot be subtracted from any points on the acceleration curve since it is a steady result and thus, strictly speaking, cannot be used to alter the computed unsteady results. It does, however, serve to indicate that at some point during both the upstroke and the downstroke, a viscous drag force of approximately this magnitude would be experienced by the wrasse. It therefore also indicates that if an unsteady NavierStokes computation were performed, one would have a substantially lower positive region, or possibly no positive region, during the downstroke.
The computed results for foreaft acceleration during the upstroke agree well with the experimental results. The experiments show an increase in forward acceleration, to a maximum occurring between 40-50% of the upstroke, then decreasing steadily to zero at approximately 90% of the upstroke. The maximum accelerations from the experiments were in the range 30-90 cm s-2. A forward acceleration maximum of approximately 45 cm s-2 was obtained at approximately 68% of the upstroke (Fig. 11A), with the acceleration decreasing steadily toward zero at the beginning of the downstroke.
Dorsoventral acceleration
We also compare our computed results for dorsoventral acceleration
(Fig. 11B) with that from
fig. 5B of Walker and Westneat
(1997). The distributions of
accelerations for fishes swimming at 47 cm s-1 rise from a negative
value at the start of abduction (downstroke) to a maximum dorsal acceleration
at about 44% abduction, decreasing to zero at about 80%, and becoming negative
through the remainder of the downstroke. The range of the maximum dorsal
acceleration is 70-100 cm s-2, and that of the ventral
accelerations at the start of abduction is 10-30 cm s-2. Our
computed results show a ventral (downward) acceleration at the start of the
downstroke, rising to a maximum dorsal acceleration at about 40% abduction,
and then decreasing to zero at about 70% abduction and steadily becoming
negative. The magnitude of the computed dorsal acceleration is approximately
80 cm s-2 and the ventral acceleration at the start of abduction is
approximately 40 cm s-2. Again there is excellent agreement with
the experimental results.
It should also be noted here that the mass of the fish we used in our computations, 100 g, was not a mass of any of the fish population making up the 47 cm s-1 data. The masses of the fish making up the 47 cm s-1 data were 24-50 g, hence we did not actually compare our computations against the data for any specific fish in the original experiments, but rather we claim that our computed fish dynamics are representative of the dynamics observed experimentally for the bird wrasse as a species. Since the unsteady computations are in good agreement with the results from the swimming experiments, we conclude that inertial effects, ignored in the quasi-steady computations, are quite significant in the thrust- and lift-generation processes.
Summary and conclusions
We computed the unsteady dynamics about a bird wrasse with flapping and
deforming pectoral fins using a new moving-mesh capability for unstructured
adaptive meshes. The unsteady computations were compared with steady state
computations, quasi-steady state computations and experimental results. We
found that the steady state computations are incapable of describing the
dynamics associated with the flapping fins. The quasi-steady state
computations, with correct incorporation of the experimental kinematics, are
useful in determining trends in force production. They do not, however,
provide accurate estimates of the magnitudes of the forces produced.
Completely unsteady computations about the deforming pectoral fins using
experimentally measured fin kinematics gave excellent agreement, both in the
time history of force production throughout the flapping strokes and also the
magnitudes of the generated forces.
We confirmed the experimental findings on the time of occurrence of the maximum thrust during adduction and maximum lift during abduction. We concluded that 3-D inertial effects are not a minor perturbation, but are critical for accurate force computations. We also observed, through flow visualization throughout the stroke cycle, only a very small flapping-induced inward and outward spanwise velocity, acting over the near-root region of the pectoral fin. We observed a large recirculation region in the junction of the pectoral fin and the body that extends to almost half the span and is present throughout the cycle of oscillation. During the downstroke, maximum thrust and lift forces occur at approximately 32% and minimum thrust is produced at 84% of the downstroke. The thrust reaches a maximum at approximately 68% of the upstroke. The velocity vectors indicate the presence of a large leading edge vortex, the shedding of a pair of counter-rotating vortices at the end of the upstroke from the distal edge, and the shedding of a trailing-edge vortex midway during the upstroke. We did not observe a spiral leading-edge vortex on the pectoral fin. The pectoral fin flapping flow was dominated by the strong axial flow, as opposed to the flows in hovering insects such as the hawkmoth, where an attached leading-edge spiral vortex has been shown to be important in high lift generation. In pectoral fin flapping associated with swimming against a strong current, the primary need for the wrasse is to attain the axial acceleration necessary for high-speed forward motion, and some vertical position changes can be tolerated. It is possible that during hovering or low-speed maneuvering, where vertical position-keeping is more important, wrasse pectoral fin flows are more like those of the hawkmoth or Drosophila.
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