Propulsive force calculations in swimming frogs II. Application of a vortex ring model to DPIV data
1 Department of Marine Biology, University of Groningen, Biological Centre,
PO Box 14, 9750 AA Haren, The Netherlands
2 Department of Biology, University of Antwerp, campus 3 eiken,
Universiteitsplein 1, B-2610 Wilrijk (Antwerpen), Belgium
* Author for correspondence (e.j.stamhuis{at}biol.rug.nl)
Accepted 19 October 2004
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Summary |
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The wake showed two vortex rings left behind by the two feet. The rings appeared to be elliptic in planform, urging for correction of the observed ring radii. The rings' long and short axes (average ratio 1.75:1) were about the same size as the length and width of the propelling frog foot and the ellipsoid mass of water accelerated with it. Average thrust forces were derived from the vortex rings, assuming all propulsive energy to be compiled in the rings. The calculated average forces (Fav=0.10±0.04 N) were in close agreement with our parallel study applying a momentumimpulse approach to water displacements during the leg extension phase.
We did not find any support for previously assumed propulsion enhancement mechanisms. The feet do not clap together at the end of the power stroke and no `wedge-action' jetting is observed. Each foot accelerates its own water mantle, ending up in a separate vortex ring without interference by the other leg.
Key words: frog, Rana esculenta, swimming, thrust force, vortex ring, DPIV
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Introduction |
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In the absence of methods to quantify water displacements and derive thrust
forces directly, several modelling approaches have been used to map
instantaneous propulsion throughout the kick-and-glide cycle in frogs and
other self-propelling aquatic animals with a similar swimming style. These
models were based on the thrust vs drag force balance that explained
the movement of the centre of mass and were expressed in more general terms
(e.g. Blake, 1981; Daniel,
1984
,
1995
) or with special
application to frog swimming (e.g. Gal and Blake,
1988a
,b
;
Nauwelaerts and Aerts, 2003
).
Thrust forces were estimated by considering the resistive forces of the flat
propulsive structure(s) as well as the added mass of the water mantle carried
along by propulsor. Refinements of the models were found in treating the
propulsive structure as an array of elements that each contribute to
propulsion autonomously (`blade element approach'), and in redefining the
added mass volume (Blake, 1979
;
Morris et al., 1985
;
Stamhuis and Videler, 1998
).
With regard to frog swimming, differences between the centre of mass
kinematics and the modelled hind limb forces were interpreted as deficits in
thrust and were explained by assuming the existence of two additional thrust
enhancing mechanisms: (1) interference between the two feet during leg
extension, resulting in larger functional foot areas, and (2) backwards
jetting of water when the two feet were forcefully brought together at the end
of the power stroke (Gal and Blake,
1988b
).
Validation of the existence of the suggested thrust and drag mechanisms and
of the hydrodynamic models used to estimate the balancing forces can only be
done by studying the flow phenomena involved in a quantitative way.
Instantaneous quantitative information with good spatial resolution of the
flow in the vicinity of swimming animals allows for analysis of the flow
phenomena present during the propulsion generation phase as well as the flow
patterns left behind in the wake (Stamhuis
and Videler, 1995). In the accompanying paper, Nauwelaerts et al.
(2005
) studied the propulsion
process and force balance of swimming green frogs Rana esculenta
during the leg extension phase, aiming at a comparison between terrestrial and
aquatic locomotion in one species. They showed that it is possible to estimate
the propulsive forces from multi-planar digital particle image velocimetry
(DPIV) recordings and were able to derive a reliable force balance throughout
the leg extension (kick) phase. Mapping of the flow during the whole kick
phase does, however, put quite a constraint on the experimental conditions,
because the frog has to swim exactly along a certain path in the illuminated
plane (Nauwelaerts et al.,
2005
). Instead, flow phenomena in the wake, which are easier to
map, may serve as propulsion estimators.
Quantitative analyses of wakes have been performed previously in flying
birds (e.g. Spedding et al.,
1984; Spedding,
1986
,
1987
;
Spedding et al., 2003
), in
undulatory swimming fish (e.g. Blickhan et
al., 1992
; Müller et al.,
1997
) and in fish with pectoral fin propulsion (e.g.
Drucker and Lauder, 1999
).
Detailed analysis of vortex rings present in the wakes allowed for the
calculation of the impulse and energy stored in the rings, which could be used
for the derivation of average propulsive force (Rayner,
1979a
,b
,c
;
Drucker and Lauder, 1999
). So
far, this approach has not been applied to frog swimming or any similar
propulsory mode.
The aim of this paper is to verify the existence of alternative
thrust-enhancing mechanisms in swimming frogs. Furthermore, the validity of a
vortex ring approach as a method to calculate propulsive forces will be
evaluated. Finally, average propulsive forces will be calculated based on
hydrodynamic parameters in the vortex wake of swimming frogs. The results will
be compared with propulsive force calculations from
instantaneousmomentum development during active propulsion in frogs
(Nauwelaerts et al.,
2005).
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Materials and methods |
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The water was seeded with neutrally buoyant Pliolite particles (BASF,
Leverkusen, Germany) with a diameter of about 25 µm. Illumination was
provided from the top or from the side by a laser light sheet of 0.2 mm
thickness produced by a red light Krypton laser (Coherent Innova K, Coherent
Lasers Inc, Santa Clara, CA, USA; =647 nm, Pmax=1
W) provided with a cylindrical lens (f=10 mm), resulting in a
vertical or horizontal light sheet. Illuminated areas of about 5 cm x5
cm in the swimming track of the frogs were imaged from the side as well as the
top view using a high-speed video camera (Redlake MASD, Inc. San Diego, CA,
USA) at 250 Hz, shuttered at 1/500 s. The camera was provided with a 5 mm
macro ring and an aspherical Canon lens (f=50 mm, F=1.3)
operated at maximum aperture. Image series including the swimming frogs during
kick and retraction phase were analysed using a momentumimpulse
approach, as presented in Nauwelaerts et al.
(2005
). Image series of the
last part of the kick phase and of the remaining wake were analysed using a
vortex-ringmomentum approach, as presented in this paper.
Flow analysis
Frame series of the illuminated particles in the wakes of the swimming
frogs were analysed pairwise using a particle image velocimetry program (SWIFT
4.0, Dutch Vision Systems, Breda, The Netherlands). Image cross-correlation
was performed with convolution filtering applying a 51x51 pixels
interrogation area size and 65% overlap, and peaks were located using a COGW
(centre of gravity weighed to grey value) method
(Stamhuis et al., 2002).
Average particle displacements in the jet and cohering vortex ring were in the
order of magnitude of 16 pixels, and the displacement detection limit
was below 0.05 pixels. Erroneous vectors and obvious outliers were removed in
a manual validation procedure and replaced using 2D spline interpolation
(Stamhuis and Videler, 1995
;
Stamhuis, 2005a
). In general,
maximally between 2 and 10 flow vectors were interpolated on a typical data
set of 400 vectors. Calibration was performed by filming a Perspex grid prior
to the experimental session.
PIV image cross correlation resulted in a velocity vector flow field of the
imaged 2D plane with vectors arranged gridwise, allowing the derivation of the
2D distribution of the vorticity and the divergence
, and the
application of a vortex-locator d (discriminant for complex
eigenvalues). These parameters aid in identifying and characterising flow
phenomena, in particular vortices and vortex rings
(Stamhuis and Videler, 1995
;
Müller et al., 1997
). The
distribution of the divergence
was used to identify out-of-plane flow,
indicating misalignment of laser sheet and jet flow direction and thereby
potential underestimation of the velocity magnitude. Image series with
substantial out-of-plane flow components were removed from the data sets. The
distributions of velocity magnitude, vorticity
and the vortex locator
d were used to identify vortices and vortex rings and estimate their
morphometric characteristics. Cubic spline interpolation and graphic (colour
coded) representation of these parameters were used to attain a higher
morphometric accuracy and facilitate interpretation of the flow fields. Only
those image sequences in which the laser sheet crossed the middle of the foot
and thereby the middle plane of the resulting vortex ring were used for
analysis, to prevent underestimation of the vortex ring momentum.
The velocity distributions of the vortex ring jets and vortices were mapped
in the direction of the jet as well as perpendicular to the jet, and compared
to expected theoretical velocity distributions
(Milne-Thomson, 1966;
Spedding et al., 1984
;
Spedding, 1987
;
Müller et al., 1997
;
Drucker and Lauder, 1999
). This
comparison served as a validity check for application of the vortex ring
momentum calculation model (Rayner,
1979a
,b
,c
;
Spedding et al., 1984
)
(Fig. 1). Eventually, the ring
momentum, derived from the circulation, is used to estimate average propulsive
force.
|
Propulsive force estimation
The circulation of the ring-shaped vortex was calculated from the
closed-loop integration of the velocity component v in the direction of
the jet main axis (Batchelor,
1967
). For a vortex ring travelling through still fluid, this can
be simplified to a line integral along the jet main axis
(Spedding et al., 1984
):
![]() | (1) |
This again simplifies in practice to a line integration from upstream of
the vortex ring where v 0 along the Y' axis to
downstream of the vortex ring where again v
0:
![]() | (2) |
The momentum I present in the vortex ring can be obtained from the
product of the fluid density , the circulation
and the ring area
calculated from its radius R:
![]() | (3) |
![]() | (4) |
The propulsion time T was derived from the propulsive phase
history as the fraction of the leg extension phase that contributes positively
to the propulsion. The extension phase could easily be determined from the leg
kinematics. The propulsive fraction appeared to be 0.75±0.05 on
average, that is 75% of the extension time (see
Nauwelaerts et al., 2005).
Depending on the level of interference of both legs during the propulsive stroke, this propulsive force has to be corrected. The level of interference as well as the presence and contribution of alternative propulsion mechanisms such as wedge action of the feet will be estimated and evaluated from the PIV flow maps.
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Results |
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|
Vortex ring morphology and model validity
In total, seven swimming sequences from two animals out of the ten
recordings that were selected based on swimming path, showed clearly
recognisable vortex rings that had just been shed from one foot or both feet.
These sequences were analysed to obtain vortex ring impulse momentum, five
sequences showing a horizontal and two a vertical cross section of the wake.
In five cases (three horizontal and two vertical), the propulsive part of one
of the feet (mainly the toes) moved properly through the laser light sheet,
giving a high probability that the ring was cross sectioned at or close to its
centre by the light sheet. The vortex rings in the vertical cross section were
difficult to map correctly, because the ring axes were hardly ever parallel to
the laser sheet, resulting in relatively high out-of-plane flow components.
This was due to the vortex ring jets left behind by the two feet being
directed backwards but somewhat towards one another, allowing proper mapping
in a horizontal plane but complicating vertical cross section mapping.
The PIV recordings in lateral view (vertical cross section) showed vortex
rings with significantly smaller diameter compared to the recordings in dorsal
view (horizontal cross section) (Table
1) indicating the rings to be elliptic immediately after shedding.
The vortex ring model assumes the rings to be circular and is quite sensitive
to variation in the ring radius, determining the ring area (see equations 3
and 4). Therefore, we calculated a corrected ring radius based on cross
sectional area R2 for a ring and
ab for
an ellipse, R being the ring radius and a and b
being the half the short and long axis of the ellipse, respectively. The ring
radius thus used for momentum calculations was
R=(ab)0.5. Because PIV analysis of the larger
horizontally sectioned rings have a higher resolution with regard to spatial
flow velocity distribution, and because of a higher probability of
out-of-plane flow and thereby underestimation of the circulation in the
vertically sectioned rings, we used the vertically sectioned rings to correct
the ring radii of the horizontally sectioned rings, which were then used to
derive impulse momentum and average propulsive forces. The measured ring radii
from the horizontal sections had to be multiplied by 0.755±0.026
(± S.D.) to yield the corrected radii
R.
|
All vortex rings used for impulse and force derivation showed the theoretically expected velocity distributions indicative for vortex rings, Fig. 3 serving as an example. In all cases, the jet velocity v along the X' axis showed a plateau-like distribution with typical zero crossings and velocity reversal in the vortices on both sides of the jet (Fig. 3A). The jet velocity u along the Y' axis showed a symmetrical profile with the peak in the middle of the ring (Fig. 3B). The velocity distribution along the Y' axis through the vortex centres showed reversal at the vortex centre; the velocity peaks at the edges of the core and decreases with distance outside of the core (Fig. 3C). These results matched the theoretical expectations, allowing momentum calculations from the vortex ring characteristics.
|
Vortex ring momentum and thrust force
The vortex ring characteristics and velocity distributions of the
horizontal cross-sections were sampled two or three times at 0.004 s interval
from subsequent frame pairs. The results are summarized in
Table 1. The circulation
estimates showed very little variation between the samples as well as between
shots and animals. The ring radii and the propulsive period time showed more
variation, both between samples and between the animals. Because ring radius
and propulsive period time appear as R2/T in the
thrust calculations, both are responsible for the variation in the calculated
thrust forces, the radius to a higher degree than the period time.
The average total thrust force produced by both kicking legs of the
swimming frogs was found to be 0.10±0.04 N. This is in close agreement
with the average thrust forces derived from the force profiles in Nauwelaerts
et al. (2005), being
0.12±0.05 N.
Most of the filmed sequences showed angles of 520° between the
v' axis (normal axis of the ring) and the mean path of motion
of the frogs. This indicates that normally more than 90% of the thrust force
favours propulsion and that lateral forces are relatively small. This is again
in good agreement with lateral force angle estimates of about 11°
(Nauwelaerts et al., 2005).
The angle of 34° found in one sequence shown in
Table 1 was among the highest
thrust angles found during all our experiments and is regarded as exceptional.
It is most probably due to some lateral displacement of the foot for
manoeuvring purposes just before ring was shed. Because all other
characteristics of this sequence were in line with expectations and other
results, there was no reason to exclude it from the data set.
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Discussion |
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An interesting parallel here is the kick phase of the human breast stroke.
A few decades ago, the preferred motion pattern for the legs was
`retract-spread-and-close', with emphasis on closing as being an important
source of thrust by stowing and squeezing water backwards (Counsilman, 1968).
This technique is now outdated although still used in recreational swimming.
The kick-phase with the feet only slightly open in a so-called `W-bearing' to
get a large area for backwards acceleration of water, is now widely accepted
as the best propulsion style for human breaststroke. This technique is often
referred to as `Whip-action' when compared to the outdated `Wedge-action'
technique (Whitten, 1994).
Swimming frogs apparently have a natural olympic swimming style, which humans
seem to approach more and more regarding the leg technique for competitive
breaststroke swimming.
Calculation of momentum from vortex rings.
The vortex rings left behind by the swimming frogs appeared to be elliptic.
Nauwelaerts et al. (2005)
showed the mass of water that moves with and is accelerated by the foot to be
ellipsoid. Apparently, this causes the ring to be elliptic immediately after
shedding, urging for correction of the ring radius to enable the application
of momentum calculations through the vortex ring model. It may be possible
that the vortex ellipse merges into a real ring shape in time. Nevertheless,
we used the ellipse shaped ring immediately after shedding for impulse
calculations because the energy in the vortex decreases rather steeply with
time due to dissipation, and may be half of the original energy within tenths
of a second (Müller et al.,
1997
; Stamhuis,
2005b
). We trust our approach of radius correction of a recorded
ring as short after shedding as possible to give the most reliable
results.
The calculated forces based on the vortex ring model were about 20% lower
than derived through an instantaneous impulsemomentum approach applied
to water dragged along by the foot during the kick
(Nauwelaerts et al., 2005).
When comparing both methods within the same kick-cycle, which was possible for
two of the recordings with two different frogs, the average forces match even
better: 0.148 N vs 0.130 N (frog 1) and 0.069 N vs 0.082 N
(frog 2) for the vortex ring vs the instantaneous impulse approach,
respectively. This illustrates that both methods mutually support one another
but also show some variation, probably due to the simplifications that were
necessary when deriving forces for a 3D flow phenomenon from planar flow
information. On average, the results for the vortex ring method were somewhat,
but not statistically, significant (t-test, P>0.2), lower
than the instantaneous impulse method, which may be explained by the fact that
a vortex ring will practically never contain all the momentum from the kicking
foot, due to losses. Some energy will be lost at the interface of the water
mass moving along with the foot during the kick and the surrounding water.
Momentum or energy calculations derived from a vortex ring are therefore bound
to underestimate the total energy that was invested to create the vortex ring.
This was also concluded for momentum and energy derivations based on vortex
rings created by a flying bird (Spedding,
1986
) and swimming fish
(Müller et al., 1997
). We
expect the losses to be less than 20% in the frog kick case, but realistic
estimates are hard to derive at this stage. Most papers in the quite large
body of literature on vortex ring phenomena examine the creation and evolution
of vortex rings originating from pipes or nozzles (e.g.
Schramm and Riethmuller, 2001
;
Shusser and Gharib, 2000
). The
creation of vortex rings by (flat) surfaces moving through free fluid,
comparable to the frog kick, and the energetics during the vortex creation
process, have had little attention so far. More theoretical as well as
formalised experimental work would increase our understanding not only of frog
swimming but also of other rowing propulsors.
In conclusion, derivation of thrust forces from the vortex rings in the wake of swimming green frogs seems a reliable way of estimating the average thrust force during the kick phase. Momentum derived from the vortex rings will be somewhat underestimated, but there is no indication that severe underestimation will take place. For reliable momentum calculations, the vortex ring should be cross-sectioned through its centre by the illumination sheet. Deviations from ring circularity should be corrected to avoid serious errors in the momentum estimations.
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Acknowledgments |
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