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First published online August 8, 2008
Journal of Experimental Biology 211, 2669-2677 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.015883
The `upstream wake' of swimming and flying animals and its correlation with propulsive efficiency
1 Bioengineering, California Institute of Technology, Pasadena, CA 91125,
USA
2 Graduate Aeronautical Laboratories, California Institute of Technology,
Pasadena, CA 91125, USA
* Author for correspondence (jfpeng{at}caltech.edu)
Accepted 18 May 2008
 |
Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Key words: wake, upstream, fluid dynamics, locomotion
|
INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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; Drucker and Lauder,
1999; Liao et al.,
2003; Wilga and Lauder,
2004; Bartol et al.,
2005; Tytell,
2007); bird, bat and insect flight
(Spedding et al., 2003;
Videler et al., 2004;
Warrick et al., 2005;
Tian et al., 2006;
Hedenstrom et al., 2007;
Maxworthy, 1979;
Ellington et al., 1996;
Dickinson et al., 1999;
Wang, 2005) and many other
modes of locomotion (Nauwelaerts et al.,
2005; Dabiri et al.,
2006) have used this wake structure to infer the physical
mechanisms governing swimming and flying. However, in each case the focus has
been primarily on flow features near the animal appendages or downstream from
the animal, based on the assumption that the upstream flow in front of the
animal is trivial.
In most studies, the wake vortex structures are identified using velocity,
vorticity or streamline plots. Using these methods, no apparent structures are
observed in the upstream flow of an animal, especially if the upstream flow is
quiescent or has uniform incoming velocity. Recent studies
(Haller, 2001;
Haller, 2002;
Shadden et al., 2005;
Shadden et al., 2006) have
introduced a new method of fluid dynamics analysis to identify more general
types of fluid structures. These coherent structures include vortices but are
more generally fluid structures that have distinct dynamics from the
surrounding fluid. The new flow analysis method is based on Lagrangian fluid
particle trajectories rather than the traditional Eulerian velocity or
vorticity plotted at a single time instance. The technique is able to locate
vortices in the downstream wake; importantly, it also indicates fluid
structures in the upstream flow. For example, coherent fluid structures are
observed upstream of a cylinder in cross-flow
(Franco et al., 2007) although
there is no upstream vorticity (the incoming flow is uniform). Upstream
coherent structures are also seen in front of a swimming jellyfish
(Shadden et al., 2006;
Peng and Dabiri, 2007).
The identification of upstream coherent structures provides researchers
with additional information regarding fluid kinematics. Previous research has
shown that these upstream structures are indicators of fluid transport
(Shadden et al., 2006;
Franco et al., 2007). For
example, only fluid inside the upstream structures is sampled by a swimming
jellyfish; therefore, only prey inside these upstream structures can be
captured by the animal.
The correlation between the upstream structures and the energetics of animal locomotion has not been investigated previously. In the present study, we use a computational model to simulate a self-propelled swimmer and identify the upstream fluid structures. A mass flow rate is then defined based on the upstream structures, and a metric for propulsive efficiency is established using the mass flow rate and the kinematics of the swimmer. We propose that, just as the downstream wake has been traditionally correlated to the forces and energetics of locomotion, the heretofore invisible `upstream wake' also exhibits dynamic significance and variation across animal species that can inform ongoing comparative biological and engineering studies of animal swimming and flying.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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(t,s) or by the lateral position y(t,s) in the
body frame of reference (Fig.
1). Three different kinematics were used in this study
(Table 1).
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|
An inviscid vortex sheet method
(Nitsche and Krasny, 1994;
Jones, 2003;
Shukla and Eldredge, 2007) was
used to solve the flow induced by the model swimmer. This method has been
validated with experiments in a number of previous studies
(Nitsche and Krasny, 1994;
Jones, 2003). The solid body
was modeled as a bound vortex sheet, and the separated shear layers were
modeled as free vortex sheets shed at the trailing edge of the swimmer. In the
numerical procedure, the bound vortex sheet attached to the swimmer was
discretized and represented as a set of vortex filaments. The position of the
bound vortex sheet was known since it coincides with that of the swimmer for
all time t. The flow separates at the trailing edge, giving rise to a
free shear layer in the flow. A time-stepping procedure was used to release
discrete vortex elements of suitable strength from the trailing edge at each
step. The unknown bound vortex sheet strength and the edge circulations were
solved at each time step by a system of equations satisfying the continuity of
the normal velocity across the swimmer, Kelvin's circulation theorem, and the
boundedness of the velocity field (Shukla
and Eldredge, 2007).
In this inviscid formulation, the hydrodynamic force acting on the swimmer
was given by the pressure difference across the flexible plate. The pressure
difference across the bound vortex sheet [p](x,t)
(`[]' indicates the discontinuity across the plate) can be expressed
(Jones, 2003) as:
 | (1) |
is the circulation, 
is the vortex sheet strength,
u is the tangential component of the average velocity at the bound
vortex sheet, 
is the tangential component of the velocity of the plate,
and x is the spatial coordinate. The net hydrodynamic force acting on
the swimmer is therefore:
 | (2) |
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|
). The computed flow is consistent with the result
of previous digital particle image velocimetry (DPIV) measurements
(Fig. 2), with strong lateral
jets present in the wake rather than downstream jets; these lateral jets are
the key characteristic of eel swimming
(Tytell and Lauder, 2004). The
self-propelled swimming speed is higher for the model versus the real
animal [1.9 body lengths s–1 (BL s–1)
versus 1.5 BL s–1], a discrepancy that can be
primarily attributed to the lack of skin friction in the vortex sheet model.
Despite this effect, the comparable flow kinematics support the notion that
the numerical method is appropriate for this proof-of-concept analysis.
FTLE calculation and LCS extraction
To reveal the upstream wake structure, we analyzed the flow using a
Lagrangian, or particle-tracking, technique. Specifically, we computed the
finite-time Lyapunov exponent (FTLE) field of the flow and identified the
Lagrangian Coherent Structure (LCS). The FTLE is defined by:
 | (3) |
(x) measures the maximum linearized growth rate of the
perturbation 
x over the interval T. In other words, it
characterizes the amount of fluid particle separation, or stretching, about
the trajectory of point x over the time interval
[t0, t0+T]. The absolute
value |T| is used instead of T in the
definition because the FTLE can be computed for T>0, indicating
fluid particle separation, or for T<0, indicating fluid particle
attraction (i.e. fluid particle separation in backward time). An illustration
is given in Fig. 3, in which
fluid particle trajectories are used to locate the boundaries of a vortex
ring. In Fig. 3A, fluid
particle pairs straddling the front vortex boundary diverge faster than any
other arbitrary pairs in backward time, indicating a larger value of
backward-time FTLE at the front boundary of the vortex ring; in
Fig. 3B, fluid particle pairs
straddling the rear vortex boundary separate faster than any other arbitrary
pairs in forward time, indicating a larger value of forward-time FTLE at the
rear boundary of the vortex ring.
|
; Shadden et al.,
2006). FTLE calculations for the flow generated by the model
swimmer were made using an in-house code (available for download from
http://dabiri.caltech.edu/software.html)
previously validated in other studies of animal locomotion and
fluid–structure interactions (Peng
and Dabiri 2007; Peng et al.,
2007; Peng and Dabiri,
2008). LCS are defined as ridges of local maxima in the FTLE
field. These ridges were extracted visually in the present case, by using an
in-house graphical user interface. The uncertainty in this method of LCS
extraction is approximately ±5%
(Franco et al., 2007). LCS
extracted from the forward-time (T>0) FTLE fields are repelling
LCS, meaning fluid particles separate away from it; LCS extracted from the
backward-time (T<0) FTLE fields are attracting LCS, meaning fluid
particles are attracted to it.
In Eqn3, The FTLE
(x) is not explicitly written as a function of the integration
time T because the length of integration time does not affect the
location of the LCS. However, longer integration time can help to more
accurately determine the LCS locations by better resolving the ridges of local
maxima in the FTLE contour plot. Fig.
4 shows the FTLE for a vortex ring calculated with increasing
integration time T. With shorter integration time, the FTLE ridges
are thick bands and the precise location of the LCS can be difficult to
determine; whereas with longer integration time, the FTLE ridges become
sharper, i.e. LCS resolve into clearly defined thin lines. The appropriate
length of integration time depends on the particular flow being analyzed, but
the `rule of thumb' regarding the integration time in any LCS analysis is that
it should be chosen to be long enough so the LCS is clearly identifiable on
the FTLE contour plot. Since the length of integration time only affects the
ease and accuracy with which the LCS are determined, it has no effect on
foregoing efficiency calculations as long as the integration time is long
enough that LCS is clearly defined. The magnitude of the integration time
|T| in the present study is four swimming cycles in
each case.
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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). As in previous studies, there is no indication of
flow structure in the region upstream of the animal from this perspective.
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|
By contrast, Fig. 6B plots
the forward-time FTLE computed by observing the behavior of the flow as it
evolves forward in time. The repelling LCS, located by the ridges' high
forward-time FTLE values in the contour plot, are also plotted. What is
immediately striking is that this flow structure extends upstream, in front of
the swimmer. In fact, the upstream extent of the repelling LCS increases as
the amount of information regarding the forward-time behavior of the flow
increases, i.e. more of the upstream repelling LCS is revealed as the fluid is
tracked over sufficiently long durations to observe its interaction with the
swimmer (Shadden et al.,
2005). Movie 3 in supplementary material shows the temporal
evolution of the FTLE fields and corresponding LCS curves. The morphology of
the upstream fluid structures is clearly observable.
Since fluid is not attracted to the repelling LCS (by definition), the upstream fluid structure, which comprises the repelling LCS, is not readily visualized using passive flow markers (i.e. dye, smoke, etc.) as is the case for the attracting LCS. In addition, since the upstream flow typically possesses a uniform or zero velocity, the repelling LCS propagates without changing its shape until after it reaches the downstream wake (Fig. 6B). By that point, the behavior of the fluid around the swimmer is dominated by the nearby attracting LCS; hence the presence of the repelling LCS is obscured. It is largely for these reasons that the upstream fluid structures have not been observed previously. In the present case, we do not rely on the aggregation of fluid to reveal the repelling LCS. Instead, it is computed based on observed fluid particle separation in the flow. To be sure, we can only visualize the upstream fluid structures by tracking them until the associated repelling LCS has interacted with the swimmer. At this time, fluid particle separation becomes most pronounced, and we can retrospectively identify the upstream fluid structure.
As mentioned previously, a physical significance of the upstream fluid structures is that it indicates the extent of the region around the swimmer that is affected by its locomotion. In fact, in the present paradigm, locomotion is essentially the process whereby a swimming or flying animal transfers fluid from the upstream fluid structure (defined by the repelling LCS) to the downstream wake (defined by the attracting LCS). To demonstrate these concepts, we computationally labeled and tracked the behavior of fluid bounded by adjacent repelling LCS structures in the upstream fluid structure. Fig. 7 indicates that the fluid in the repelling LCS is indeed the source of fluid that comprises the subsequent downstream vortex wake (see also Movie 4 in supplementary material).
|
). Despite the fact that the parcels in
Fig. 8C cover the same amount
of upstream area as each of the repelling LCS labeled in
Fig. 7, they do not indicate
the full extent of the downstream wake. This illustrates that both the area
and the shape of the upstream fluid structures are important for capturing the
subsequent downstream wake and locomotive dynamics.
|
 | (4) |
). This does not mean that the net hydrodynamic force is
zero instantaneously. The force is typically oscillatory, resulting in
periodic acceleration and deceleration of the swimmer, as shown in
Fig. 9. This leads to `temporal
separation' of thrust and drag and a corresponding, measurable change in
momentum flux in the wake (Tytell,
2007). We wish to define an efficiency metric that utilizes the
upstream fluid structures to quantify this effect.
|
 | (5) |
is the fluid density, 
is the average forward
velocity over a stroke cycle, and 
is
the time-average of the width (peak-to-peak in the lateral direction) of the
LCS w(x) (see Fig.
7A). The width w(x) is swept out by the upstream
wake as it propagates downstream. Given the mass flow rate, the net change in
the momentum flux due to periodic acceleration and deceleration of the swimmer
can be expressed as 
U,
the product of the mass flow rate and the variation (maximum minus minimum) of
swimming velocity of the swimmer over a stroke cycle.
Using the momentum flux as a scale for the thrust, a metric for efficiency
is introduced as:
 | (6) |
U=0), there would be no net momentum flux. However, in
reality, the forward velocity is only quasi-steady since the reciprocal motion
of the appendages causes temporal variations in swimming velocity
(Daniel, 1984).
The efficiency of the self-propelled swimmer using each of the kinematics
in Table 1 was calculated after
the swimmer reached a steady mean velocity
(Fig. 9B). The efficiencies are
plotted in Fig. 10 against
Strouhal number (St=fA/, where f is
the tail beat frequency, A is the peak-to-peak trailing edge
excursion, and is the mean swimming velocity). For each
swimming kinematics, the efficiency has a peak, located at St=0.23,
0.18 and 0.27 for kinematics 1, 2 and 3, respectively. This is consistent with
previous studies of oscillating foils and flying/swimming animals that
indicate optimal propulsive efficiency at Strouhal numbers within the range of
0.2 to 0.4 (Taylor et al.,
2003). The efficiency is similar for kinematics 1 and 2 but lower
than that of kinematics 3, indicating that kinematics 3 is the best of the
three in terms of swimming efficiency. The reason that kinematics 3 has the
highest efficiency of the three is that it requires less total power than
kinematics 1 and swims faster than kinematics 2.
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|
DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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;
Peng and Dabiri, 2007)
indicate the existence of upstream fluid structures in that flow as well,
albeit with the possibility that the entire structure was not captured in the
measurement. By following fluid particles, it is shown that – similar to
the model swimmer – only the fluid inside these upstream structures
interacts with the animal (see Movie 7 in supplementary material). Therefore,
despite the simplicity of the computational model used here, we hypothesize
that the upstream fluid structure is a generic and morphologically diverse
feature of locomotion in real fluids.
The upstream fluid structures provide a new focus for fluid dynamic studies
of swimming and flying. The upstream structure indicates the portion of fluid
that interacts with the animal, thus enabling definition of a mass flow rate
induced by locomotion. The new metric for efficiency (
LCS),
which is based on change in the momentum flux due to the periodical
acceleration of center-of-mass, can be used as a new metric for evaluating
swimming performance.
The mass flow rate and the new
metric for efficiency LCS are calculated from the width of the
fluid structure w. The width of the fluid structure is larger than
the flapping amplitude at the trailing edge, indicating that a larger region
of fluid interacts with the swimmer. Fig.
11 also plots an efficiency metric similar to that in
Eqn6 but with the upstream fluid
structure width w replaced by the flapping amplitude A. The
efficiency based on A does not correlate well with the efficiency
based on w. This is because w depends not only on A
but also on U. To show this, w was calculated for swimming
at a constant velocity (2BLs–1) over a range of tail
amplitudes and for swimming over a range of velocities with a fixed flapping
amplitude (of 1.44BL). The results are plotted in
Fig. 12. The width of the
upstream fluid structure scales both with increasing A and with
increasing U. Therefore, A provides less information about
the locomotion than w, and it cannot take the place of w in
the analysis.
|
|
Another potentially interesting application of fluid structure shown in
this study is to calculate the Strouhal number based on w (as
St=fw/), which is similar to its classic
definition based on the width of the wake for vortex shedding by bluff bodies
(e.g. Triantafyllow et al., 1991) rather than the flapping amplitude used in
most animal swimming and flying studies. The efficiencies in
Fig. 10 are plotted against
the modified St in Fig.
13. Because w is larger than A, all three curves
shift to the right, with the new peaks located at 1.00, 0.92 and 0.91 for
kinematics 1, 2 and 3, respectively. Interestingly, the new peak efficiency
for all three kinematics is more tightly constrained using the modified
St definition. This result should be investigated further, especially
in light of previous studies indicating St tuning for a broad range
of swimming and flying animals (Taylor et
al., 2003).
|
and U) are
required for the measurement. To compare the efficiency of animals or
propulsion systems for which the mechanical power input is the same, only the
numerator of Eqn 6 is needed.
Where needed, the mechanical power 
may be determined by using existing physiological or mechanical measurement
techniques (e.g. Biewener,
2003; Krueger,
2006). Alternatively, the efficiency can be approximated by the
ratio:
 | (7) |
The analytical framework developed currently was demonstrated on a model
swimmer that does not include the effects of viscous drag. However, the
efficiency metric is only affected insofar as the viscous drag contributes to
the total mechanical power in the
efficiency calculation. The mass flow rate in real flows can still be
determined without loss of generality using the methods described here. Where
viscous losses are neglected, the proposed efficiency metric will overestimate
the true performance (i.e. since the denominator of the efficiency will be
underestimated). In theory, the proposed measure is most accurate when the
power loss due to friction, Pfric, is small relative to
the total mechanical power,
=Pprop+Pwake+Pfric,
where Pprop and Pwake are useful power
for propulsion and wake power, respectively. Equivalently, we require that the
Reynolds number, Re, is
aU3/(Pprop+Pwake),
where a is the wetted surface area of the swimmer. The Reynolds
number relationship reflects the relative importance of inertial effects
(embodied in the aU3 term) and viscous effects
(embodied in Pfric).
When applying the proposed methods to particle image velocimetry (PIV)
measurements, the field of view should be large enough to cover some distance
upstream of the animal. For animals with periodic stroke patterns, the LCS is
also periodic with the same frequency (or an integer multiple thereof). Hence,
the distance upstream of the animal should be equal to or larger than the
distance the animal can swim during a stroke cycle, i.e. swimming velocity
multiplied by stroke period. The proposed method is also robust to noise in
PIV measurements. It has been demonstrated that large velocity errors still
preserve reliable predictions on Lagrangian coherent structures, as long as
the errors remain small in a special time-weighted norm
(Haller, 2002).
Although the analytical framework was demonstrated in a two-dimensional flow, the analysis can be extended to three-dimensional flows, in which the LCS are surfaces rather than curves. Although volumetric flow measurements are ideal for this purpose, two-dimensional PIV data can also be utilized by collecting measurements on multiple parallel planes. The mass flux per unit depth in each plane can then be summed to determine the total mass flux. A potential advantage of the present methods is that, given the fact that Eulerian PIV is more difficult than Lagrangian particle tracking velocimetry in three-dimensional flows, the present analysis would be especially well suited to three-dimensional experiments since LCS can be directly calculated from particle trajectories.
In summary, the upstream fluid structures visualized by the LCS analysis provide new information regarding the interaction between swimming/flying animals and the fluid environment. These animal–fluid interactions have broad consequences for predation, reproduction and other behavioral functions. As previous measurements of animal swimming and flying are re-examined and new observations are made using the methods described here, the upstream fluid structures can become as useful as the downstream wake has traditionally been for comparative biological and engineering studies of animal locomotion.
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Acknowledgments |
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|
Footnotes |
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