Propulsive performance of unsteady tandem hydrofoils in a side-by-side configuration
Peter A. Dewey, Daniel B. Quinn, Birgitt M. Boschitsch, and Alexander J. Smits
Citation: Physics of Fluids (1994-present) 26, 041903 (2014); doi: 10.1063/1.4871024
View online: http://dx.doi.org/10.1063/1.4871024
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PHYSICS OF FLUIDS 26, 041903 (2014)
Propulsive performance of unsteady tandem hydrofoils
in a side-by-side configuration
Peter A. Dewey,1,a) Daniel B. Quinn,1 Birgitt M. Boschitsch,1
and Alexander J. Smits1,2
1
Department of Mechanical and Aerospace Engineering, Princeton University,
Princeton, New Jersey 08544, USA
2
Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800,
Australia
(Received 11 December 2013; accepted 31 March 2014; published online 24 April 2014)
Experimental and analytical results are presented on two identical bio-inspired hydrofoils oscillating in a side-by-side configuration. The time-averaged thrust production
and power input to the fluid are found to depend on both the oscillation phase differential and the transverse spacing between the foils. For in-phase oscillations, the
foils exhibit an enhanced propulsive efficiency at the cost of a reduction in thrust.
For out-of-phase oscillations, the foils exhibit enhanced thrust with no observable
change in the propulsive efficiency. For oscillations at intermediate phase differentials, one of the foils experiences a thrust and efficiency enhancement while the other
experiences a reduction in thrust and efficiency. Flow visualizations reveal how the
wake interactions lead to the variations in propulsive performance. Vortices shed
into the wake from the tandem foils form vortex pairs rather than vortex streets. For
in-phase oscillation, the vortex pairs induce a momentum jet that angles towards the
centerplane between the foils, while out-of-phase oscillations produce vortex pairs
C 2014 AIP Publishing LLC.
that angle away from the centerplane between the foils. °
[http://dx.doi.org/10.1063/1.4871024]
I. INTRODUCTION
The locomotion of animals and fish through a fluid medium has received significant attention
in recent years, particularly in the context of developing novel bio-inspired autonomous underwater
vehicles.1–4 These studies typically simplify the problem of bio-inspired propulsion to examine
the performance of an isolated propulsive surface oscillating in an oncoming flow, and to this end
significant progress has been made.5–7 However, the propulsive surfaces used by fish are seldom used
in isolation. They often interact with solid surfaces to generate an unsteady ground effect,8–10 or with
other fish in the form of schooling,11 and multiple interactions may occur among the appendages
and fins on the animal itself.12–15 While there are likely many evolutionary reasons to explain
these behaviors that extend beyond hydrodynamic implications,11, 16–18 it has been shown by Akhtar
et al.15 and Boschitsch, Dewey, and Smits19 that fish swimming in an in-line configuration can
obtain a hydrodynamic benefit, which may have important implications for the design of underwater
vehicles.
The case of foils actively swimming in a side-by-side configuration has also received some
limited attention.20 One of the first investigations into the hydrodynamics of fish swimming in pairs
(or schools) was conducted by Weihs.21, 22 Using an inviscid potential flow model, Weihs suggested
that schooling fish could significantly enhance their thrust production. Dong and Lu23 computed
the flow over wavy foils traveling in a side-by-side arrangement and found a reduction in power
consumption when the foils oscillated in-phase with one another, and an enhancement in the fluid
forces when the foils oscillated out-of-phase with one another. No other oscillation phase differentials
a) [email protected]
1070-6631/2014/26(4)/041903/16/$30.00
26, 041903-1
°
C 2014 AIP Publishing LLC
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wake
ground, y = 0
wake image
FIG. 1. Illustrating the method of images applied by Quinn et al.10 to study an unsteady foil in ground effect. In this manner,
an unsteady foil operating near a solid surface is analogous to a pair of foils oscillating out-of-phase with one another.
were considered in this study. In addition, Wang and Russell24 computationally examined forewing
and hindwing interactions in hovering dragonflies and noted that an out-of-phase motion between
the wings was better suited for hovering motions, while in-phase motion was better suited for takeoff
maneuvers.
More recently, Blevins and Lauder9 and Quinn et al.10 examined the hydrodynamics of an
unsteady foil operating near the ground. Only a single foil was studied, but by invoking the method
of images the ground can be represented in an inviscid sense by a second foil oscillating out-ofphase with the original (see Figure 1). The method of images ensures a no-flux condition through
the ground at the expense of allowing for a slip boundary condition along the ground. The case of an
unsteady propulsor in ground effect is therefore analogous to a pair of foils oscillating out-of-phase
with one another. Quinn et al.10 found that the time-averaged thrust produced by the foil increased
monotonically as the distance to the ground decreased while the propulsive efficiency of the system
stayed relatively constant, indicating that swimming near the ground can enhance thrust up to 70%
at little extra energy cost. Furthermore, the time-averaged wake of unsteady foils in ground effect
was found to angle away from the wall due to the formation of vortex pairs in the wake.10
Here, we examine the propulsive characteristics and wake structures of a pair of foils oscillating
in a side-by-side configuration using experimental and analytic techniques. We are motivated by
previous studies that have found that propulsive benefits are achievable for tandem foils; however,
we note that many questions remain unanswered in the literature for this field. Most previous studies
in the field focus on a very narrow parameter space that consider only in-phase or out-of-phase
oscillations for a fixed spacing between the foils. The conclusions of such works, while helpful, are
limited to a small subset of the parameter space. The nature of the propulsive characteristics and wake
dynamics of tandem foils remain unclear when both the oscillation phase differential and spacing
between the foils is systematically varied. To better our understanding of the underlying physical
mechanisms governing the propulsion of tandem foils, we focus our efforts on three questions in
particular: (1) How do the propulsive characteristics vary as a function of oscillation phase differential
between the foils? (2) How do the propulsive characteristics vary as a function of transverse spacing
between the foils? (3) What are the wake dynamics of tandem foils and can they be simply modeled?
By focusing our efforts at bettering understanding the underlying physical mechanisms, we hope
that this work will aid biologists and engineers alike. In particular we call attention to two areas
for potential application: the development of bio-inspired autonomous underwater vehicles, and the
development of energy harvesting systems based on the unsteady oscillations of foils due to fluid
structure interactions. In the latter case, novel flow energy harvesters, such as the ones described
by McKinney and DeLaurier,25 Zhu, Haase, and Wu,26 and Boragno, Festa, and Mazzino,27 extract
energy from a moving fluid through passive oscillations (that is, without external actuation). We
hope that the forces and wake interactions described in the present work can better inform the design
of these systems when multiple energy harvesters are placed side-by-side in arrays.
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b = 0.1C
U∞
y
d
x
A
C
FIG. 2. Foil cross-sections showing the chord length C, the thickness b, spanwise spacing d, and peak to peak amplitude of
motion of the trailing edge A. The foils have identical geometries and amplitudes of motion. The dashed lines correspond to
the envelope of motion for the foils.
II. EXPERIMENTAL METHODS
The experimental procedures used for the current effort are similar to those used by Dewey
et al.28 in a study of a single flexible pitching panel, Quinn et al.,10 in a study of a single foil
oscillating in ground effect, and Boschitsch, Dewey, and Smits19 in a study of two foils in an in-line
configuration.
The two hydrofoils were constructed of anodized aluminum having a chord length C = 79.4 mm
and a span S = 280 mm. The cross-section of each foil has a semicircular leading edge with a
maximum thickness of b/C = 0.1 that tapers along straight lines to the trailing edge, as shown in
Figure 2. The foils spanned the entire depth of the water channel to mitigate three-dimensional
effects, and were aligned with the free-stream velocity so that the average angle of attack over an
oscillation period was zero. A closed-loop, free-surface water channel with a test section measuring
0.46 m wide, 0.29 m deep, and 2.44 m long was used for the experiments (see Figure 3). The
free-stream flow speed (U∞ = 0.06 m/s) and chord based Reynolds number (Re = 4700) were the
same as in the work of Quinn et al.10 and Boschitsch, Dewey, and Smits,19 and were inspired by the
swimming of the bluegill sunfish.15
load cell
fulcrum
lever arm
encoder
servo motor
U∞
mirror
camera
torque sensor
air bearing
hydrofoil
FIG. 3. Experimental setup. The inset depicts the actuation mechanism and lever arm used for force measurements for one
of the foils. The setup for the other foil was identical.
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Two independent actuation mechanisms were used to pitch the foils sinusoidally about their
leading edges so that the angular displacement of foil 1 is given by θ1 = θ0 sin(2π t ∗ + φ), and for
foil 2 by θ2 = θ0 sin(2π t ∗ ). Here, θ 0 is the maximum pitching angle, t∗ = ft, f is the oscillation
frequency, t is the time, and φ is the phase difference in actuation. The origin of the x-y coordinate
system is aligned with the trailing edges of the foils and located at the centerline between the foils,
as shown in Figure 2. Each actuation mechanism was supported by a frictionless air-bearing system
and abutted against a lever arm applying a force to an Omega LCAE-600 load cell. In this way,
the net thrust T of each foil could be measured for all cases considered. The power input to the
fluid P was determined for each foil by measuring the reactionary torque applied by the motor and
the instantaneous angular velocity of the foil using a digital encoder. The power input to the fluid
was determined by measuring the power required to oscillate the foil in the moving fluid and then
subtracting the power required to operate the mechanism with the foil removed. Only cycle averaged
force values were determined.28, 29 We also do not consider the lift, or lateral forces generated by the
oscillating foils.
Particle Image Velocimetry (PIV) was used to examine the wake of the foils. The flow was
illuminated from a light sheet generated by a 50 mJ/pulse Nd:YAG laser (Litron Nano L 50-50) and
image pairs were acquired using an Imager sCMOS camera (2560 × 2160 pixels). The system was
triggered by a programmable timing unit with a 10 ns jitter. The flow was seeded with neutrally
buoyant ceramic spheres with a mean diameter of 10 µm. Five pass windows in DaVis 8.1.4 software
were used, with a final window size of 16 × 16 pixels with 50% overlap, resulting in a field of 320
× 270 vectors, which corresponds to approximately 135 vectors per chord. The “instantaneous”
fields were acquired by phase-averaging over 4 discrete phases per oscillation period, while the
“time-averaged” flow fields were determined by averaging data acquired at 20 discrete phases per
oscillation period over 10 oscillation periods. These techniques are described in more detail by
Dewey, Carriou, and Smits.30
III. ANALYTIC METHODS
Naguib, Vitek, and Koochesfahani31 developed a vortex array model to simulate the wake
behind an oscillating propulsor using a train of finite-core vortices with alternating sign. The model
gives the unsteady velocity field by computing the induced velocities according to the Biot-Savart
law, and it successfully reconstructed the time-averaged wakes observed in the experiments of
Koochesfahani.32 This model assumes a “frozen” wake where the mutual interactions of the wake
vortices are neglected, but since the wakes considered here are potentially asymmetric, we will
construct a more general form of this vortex array model to include mutual vortex interactions in the
wake.
The notation used here is shown in Figure 4. We introduce the complex velocity ̟ (z) = u − iv,
where u and v are flow velocities in the x- and y-directions and the complex coordinate z = x +
iy. Hence we define the dimensionless quantities ̟ ∗ ≡ ̟ /U∞ , z∗ = z/C, Ŵ ∗ = Ŵ/(U∞ C), and
t∗ ≡ ft, where Ŵ is the circulation. Following Naguib, Vitek, and Koochesfahani,31 we model each
vortex n
z − ζn
v
u
∗ . The
FIG. 4. Vortex array model notation (φ = 0). The dimensionless time at which vortices were introduced is given as tshed
complex velocity ̟ at point P results from the free-stream velocity U∞ and the induced velocity from other wake vortices,
including the highlighted vortex n whose center is located at ζ n .
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wake vortex as a Gaussian distribution of vorticity such that its circulation profile is given by
·
´2 ¸
³
|
− |z−ζ
R
,
Ŵ ∗ (z ∗ ) = Ŵ0∗ 1 − e
where Ŵ0∗ is the vortex circulation as |z∗ | → ∞, R is the vortex core radius, and ζ is the complex
coordinate of the vortex center. The parameter R can be thought of as a smoothing parameter to
minimize singularities and numerical noise, and was fixed at 0.3C for all data shown. The far-field
circulation Ŵ 0 was assumed to scale with 2π 2 St2 based on the bound circulation predicted by the
Theodorsen33 model. Here, St is the Strouhal number defined by fA/U∞ .
To include mutual vortex interactions in the wake, we introduce four vortices per oscillation
cycle beginning at t∗ = 0 (two for each foil). We recognize that it is possible for more than two
vortices to be shed per oscillation period from each foil,30, 34 but for simplicity we restrict our model
in this way. The vortices alternate in sign, with positive vortices created at z = (± d + A)i/2 and
negative vortices created at z = (± d − A)i/2. They are emitted into the wake at the time when
the foils reverse direction. Figure 4 highlights an arbitrary vortex n and point P to illustrate the
definitions used here.
The complex velocity induced by N vortices and the uniform flow is given by
̟ ∗ (z ∗ ) = 1 +
N
∗
i X Ŵ j (z ∗ )
,
2π j=1 ζ j∗ − z ∗
(1)
where Ŵ ∗j and ζ j∗ are the strength and position of the jth vortex, respectively. At each time step,
the vortices advect via Routh’s rule,35 that is, they move with the local velocity induced by all
flow elements besides themselves. Vortex 3, for example, moves a distance ̟ ∗ (ζ3∗ )1t ∗ every time
step, where ̟ ∗ is evaluated for all j 6= 3. A time step of 1t∗ = 0.001 was chosen such that the
time-averaged velocity magnitudes differed by an average of only 1% when 1t∗ was halved. To find
the time-averaged velocity, Eq. (1) was evaluated at each point on a 60 × 40 grid extending from
x/C = 0 → 4 and y/C = −1.5 → 1.5. To minimize transient effects on startup, time-averaging began
after 8 oscillation cycles and continued until instantaneous velocities on the grid differed by no more
than 1% between cycles.
IV. PARAMETER SPACE
The cycle averaged net thrust (T̄ ) and cycle averaged power input to the fluid (P̄) were measured
for a range of foil spacings D∗ and oscillation phase differentials between the foils φ. The thrust and
power are typically given in non-dimensional form by the coefficients of thrust CT and power, CP ,
where
CT =
T̄
1
2
ρU∞
SC
2
and
CP =
P̄
1
3
ρU∞
SC
2
.
(2)
The propulsive efficiency is then defined as the Froude efficiency given by
η=
T̄ U∞
CT
,
=
CP
P̄
(3)
which represents the fraction of total energy in the wake that results in useful energy output for the
propulsor.
To compare the thrust, power, and efficiency data for the foils in the side-by-side configuration
with the case of an isolated foil, we use an asterisk to denote normalized values so that, for example,
C T∗ = C T /C T,s represents the ratio of the coefficient of thrust for a side-by-side foil to that of an
isolated foil. Figure 5 shows the thrust and efficiency data for the isolated foil as a function of
St = fA/U∞ . We note that the foil transitions from net drag to net thrust at St ≈ 0.13, and that a peak
in propulsive efficiency occurs at St = 0.25.
For the tandem foil configuration, thrust, power, and propulsive efficiency were measured for
foil spacings from D∗ = d/C = 0.5 to 2 (in steps of D∗ = 0.25) and oscillation phase differences
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1
0.25
0.8
0.2
0.6
CT ,s
0.15
ηs
0.4
0.1
0.2
0.05
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
St
St
(a)
(b)
0.4
0.5
FIG. 5. Propulsive performance of an isolated foil shown as a function of Strouhal number: (a) Coefficient of thrust and (b)
propulsive efficiency. The subscript s denotes that these are the single foil results.
from φ = 0 to 2π (in steps of π /8). All data were acquired at a Strouhal number of St = 0.25,
which corresponds to a reduced frequency of k = 2π fC/U∞ = 2π , a value that is consistent with
observations of animals in nature,36 and equal to the value that optimized the propulsive efficiency
of an isolated foil for the same configuration (Figure 5). We leave the drag-to-thrust transition case
for the tandem foils for future work, and instead focus on the Strouhal number that optimizes the
propulsive efficiency for the single foil case. The full data set yields 119 separate cases and represents
a significantly broader and more refined parameter space than has previously been considered.23
The maximum foil spacing was limited to D∗ = 2 to avoid interference from the walls of the test
section,10 and the minimum spacing was limited to 0.5 because of interference between the actuation
mechanisms. For convenience, the data corresponding to the isolated foil case, and their associated
uncertainty, are summarized in Table I for the Strouhal number and Reynolds number of interest.
In general, the uncertainty of the measurements for the thrust coefficients, power coefficients, and
propulsive efficiency are ±13%, ±9%, and ±18% of their stated values, respectively.
V. PROPULSIVE PERFORMANCE
Figure 6 shows the coefficients of thrust and power as a function of phase for each foil, and for
the sum of the two foils, at D∗ = 0.5. The results depend strongly on phase. As the foils oscillate
in-phase (φ = 0 and 2π ), the coefficients of thrust and power are reduced to approximately half
of that of the isolated foil. As the phase differential is increased (φ = π /2), the foil leading in the
oscillation period (foil 1) yields an enhanced thrust while the foil trailing in the oscillation period
(foil 2) yields less thrust compared to an isolated foil (C T∗ = 1.4 and 0.6, respectively). At φ = π /2
the combined thrust of the two foils is approximately twice that of an isolated foil, so there is no net
benefit to the system, although one foil clearly generates more thrust than the other. As φ → π , the
individual foils are close to a local peak in performance, and at φ = π a thrust enhancement of about
70% is observed for the system compared to an isolated foil, which is comparable to the maximum
benefit found by Quinn et al.10 for a foil oscillating in ground effect. As the phase increases beyond
π , the trends are reversed because of symmetry.
The results offer new insight into possible actuation strategies for tandem propulsors. Consider
a single agent employing two fins, such as a fish or underwater vehicle, or two agents operating
TABLE I. The performance parameters and uncertainty of a single foil
pitching in isolation at St = 0.25 and Re = 4700. These Strouhal number
and Reynolds number are used for all experiments reported here.
CT, s
CP, s
ηs
0.15 ± 0.02
0.66 ± 0.06
0.22 ± 0.04
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(a)
(b)
FIG. 6. Propulsive performance as a function of phase differential φ for D∗ = 0.5. (a) Coefficient of thrust for foil 1, foil 2,
and the sum of the foils 1 and 2. (b) Coefficient of power for foil 1, foil 2, and the sum of foils 1 and 2. Note that the horizontal
lines at C T∗ , C ∗P = 1 and 2 represent the performance of a single isolated foil and the sum of two isolated foils, respectively.
independently of each other, such as two fish in a school. In either case, if the aim were to minimize
power consumption, the system is best served by oscillating in-phase (φ = 0), but if the aim were
to maximize thrust production, the system is best served by oscillating out-of-phase (φ = π ). The
case where φ = π /2 is more complex because the foils will produce different amounts of thrust,
and the strategy selected by a given propulsor will likely depend on the objective. For instance, if
two fins are operating on a single agent, an intermediate phase differential could yield enhanced
maneuverability due to the moment generated by the imbalanced thrust production. If instead the
propulsors are independent of one another and instead competing (for instance, a predator-prey or
feeding scenario), each should attempt to be ahead in the flapping cycle to enhance their own thrust
while lessening the thrust produced by their competitor.
To help understand the thrust response of each foil, consider the velocities induced by each foil
during an oscillation cycle. When φ = π , foil 1 induces lateral velocities that increase the effective
angle of attack of foil 2. In the same way, foil 2 increases the effective angle of attack of foil 1. This
mutual interaction increases the bound circulation of the foils throughout the flapping cycle, thus
increasing their thrust production. When φ = 0, however, the opposite effect occurs: foil 2 induces a
flow that decreases the effective lateral velocities acting on foil 1 (and vice versa), and the circulation
and thrust of the individual foils and the combined system decreases.
When φ = π /2 and φ = 3π /2, the lateral foil velocities are in the opposite direction during parts
of the cycle, but in the same direction for other parts. Figure 7 shows the pitch angle of the leading
(L) and trailing (T) foils through one cycle for φ = π /2. The portions of the cycle where the two foils
have opposing velocities are shaded gray, and the portions where vortex cores are expected to be
shed from each foil (according to Refs. 10, 19, 30, 37, and 38) have been highlighted by thickening
the pitch angle curve. Previous work has shown that for a rigid flapping foil, the majority of the
thrust is produced as vortices are shed into the wake of the foil.23, 39 Note that for the foil leading in
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L
L
L
L
T
T
T
T
L
T
FIG. 7. Top row: schematic of the two foils and the wake vortices through the oscillation cycle. Bottom row: pitch angle θ
for the leading foil (L) and trailing foil (T) through the oscillation cycle. Portions of the cycle when vorticity is shed into the
wake are denoted by a thickened line. Portions of the cycle when the two foils are moving in opposite directions are denoted
by a shaded gray background.
phase, the bulk of the thrust production occurs when the foils are moving towards each other, while
the reverse is true for the foil trailing in phase. The circulation will be enhanced for the leading foil,
and diminished for the trailing foil, to yield a differing thrust response for each of the foils. As will
be shown in Sec. VI, the vortices shed by the foil leading in the oscillation period do in fact contain
more circulation in comparison to the vortices shed by the foil trailing in the oscillation period.
The thrust and power coefficients as a function of foil spacing are shown in Figure 8 for a number
of phase differentials. The data for only one of the foils are shown here for brevity, but the general
trends we observed were consistent for both foils. As seen in Figure 8, the interactions between
the foils decrease as the foil spacing increases. In particular, the upper and lower limit containing
the data follows a power law decay given by D∗−0.4 , which is the same scaling proposed by Quinn
et al.10 for foils in ground effect. As expected, the power law decay is centered about C T∗ , C ∗P = 1,
which indicates that as the foils become further apart they begin to act as independent foils. We also
observe that at each foil spacing the trends shown in Figure 6 hold true. That is, in-phase oscillations
consistently lead to a reduction in thrust and power, out-of-phase oscillations consistently lead to
an enhancement in thrust and power, and the intermediate phase differentials lie in between these
limits.
In summary, three principal observations can be made in comparing the performance of the
system to the performance of an isolated foil: (1) in-phase oscillation reduces the thrust produced
FIG. 8. Propulsive performance for foil 1 as a function of foil spacing D∗ . (a) Coefficient of thrust and (b) coefficient of
power. The data are contained in the envelop defined by the dashed lines, denoting a power law with a decay ∼D∗−0.4 . The
data for the various phase differentials are denoted by the solid lines, note that the line colors defined in the legend are used
in both figures.
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FIG. 9. Propulsive efficiency as a function of phase differential φ for D∗ = 0.5 for foil 1, foil 2, and the sum of foils 1 and
2. The horizontal lines at η∗ = 1 and 2 represent the performance of a single isolated foil and the sum of two isolated foils,
respectively.
and power consumed, (2) out-of-phase oscillation enhances the thrust produced and power consumed,
and (3) oscillation at intermediate phases (φ = π /2 and 3π /2) enhances the thrust produced and
power consumed by one of the foils and reduces them for the other foil. In all cases, an increase
(or decrease) in the coefficient of thrust was accompanied by a subsequent increase (or decrease) to
the coefficient of power. The magnitude of the modification to the coefficient of thrust and power
decays as the spacing between the foils is increased.
The efficiency results are shown in Figure 9 for D∗ = 0.5. The total efficiency of the system,
represented by the summation of foils 1 and 2, does not significantly deviate from the efficiency of
the isolated foils for most phase differences. The exception is the in-phase case (φ = 0), where the
performance of the system is significantly (35%) better, which is consistent with the observations
by Dong and Lu23 with respect to wavy foils. It is important to note that the relative gain in
efficiency comes with a significant loss in thrust (Figure 6). For the out-of-phase condition (φ = π )
the efficiency of the system remains unchanged from the efficiency of an isolated foil, within the
experimental uncertainty, although the foils generate significantly more thrust (Figure 6(a)). This
observation agrees with the findings of Quinn et al.10 who examined unsteady foils in ground effect.
For the intermediate case (φ = π /2), the net performance of the total system remains unchanged in
comparison to isolated foils, though the efficiencies of each foil is affected. The leading foil (foil 1)
demonstrates an enhanced propulsive efficiency, while the efficiency of the lagging foil (foil 2)
decreases.
As noted earlier, the actuation strategy selected by a given agent will depend on its objectives.
Propulsors working together can enhance their efficiency by oscillating in-phase at the cost of
decreased thrust production (Figure 6), and they can enhance thrust production without sacrificing
efficiency by oscillating out-of-phase. For two competing agents, either agent can achieve a boost
in propulsive efficiency while simultaneously reducing the efficiency of the other agent by leading
in the oscillation cycle.
VI. WAKE DYNAMICS
Hydrogen bubble flow visualizations for foils operating in-phase and out-of-phase are displayed
in Figures 10 and 11, respectively. Since the visualizations were difficult to conduct at high Strouhal
number, the Strouhal number was limited to 0.125, compared to 0.25 for the propulsive performance
measurements. The visualizations remain instructive.
In the case where the foils are pitching in-phase (Figure 10), the two foils simultaneously
generate a pair of parallel like-signed shear layers that advect into the wake of the foils. Over the
course of a half oscillation period, the shear layers roll-up into a series of like-signed vortices.
When the foils reach their minimum (or maximum) positions, they change direction and another pair
of like-signed vortices forms that has the opposite sense of rotation in comparison to the vortices
generated in the previous half-oscillation period. The wake generated from each foil is markedly
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FIG. 10. Hydrogen bubble visualization for D∗ = 1, φ = 0, and St = 0.125. Flow is from left to right.
different from the reverse von Kármán vortex street commonly observed in the wake of single
oscillating foils,34, 38 which is comprised of two vortices shed per oscillation period. The presence
of a second foil oscillating in-phase appears to excite a secondary instability that results in several
vortices being formed per oscillation period.
In the case where the foils are pitching out-of-phase, one period of oscillation generates two
sets of vortex pairs (Figure 11). The first vortex pair is generated when the foils are pitching away
from the centerplane. As the foils sweep from the centermost position outward, a reverse flow occurs
in the region between the foils that induces the formation of a single vortex core at the trailing
edge of each foil. As the foils reach their outermost position and change direction, they both pitch
FIG. 11. Hydrogen bubble visualization for D∗ = 1, φ = π , and St = 0.125. Flow is from left to right.
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FIG. 12. Wake structures for D∗ = 1, φ = 0, St = 0.25. Instantaneous vorticity fields for (a) vortex array model and (c)
PIV experiments. Time-averaged velocity fields for (b) vortex array model and (d) PIV experiments. The letter label gives
the sequence of vortex shedding, with A being the one released most recently, and subscripts 1 and 2 indicate the foil from
which the vortex originates.
towards the centerplane and induce a downstream flow that exceeds the free-stream velocity. A
shear layer is generated that creates a second vortex pair that is of opposite sign to the first vortex
pair. The resulting structure is thus comprised of a vortex quadrupole, where a saddle point is
located at the center of the quadrupole. Subsequent vortices shed by the same foil are connected by a
continuous shear layer, indicating that the secondary instability that was observed in the in-phase case
(Figure 10) is not present in the out-of-phase case.
We now connect these observations to the results obtained using the vortex array model and
the PIV experiments. We note that the vortex array model is employed here so that a qualitative
comparison can be made with the experiments. We are not seeking to directly compare the quantitative
results between the vortex array model and the PIV, but rather use the vortex array model as a
tool to understand if, and to what extent, the vortex dynamics can be described using an inviscid
argument.
The instantaneous vorticity and time-averaged velocity fields for the in-phase case (φ = 0)
are shown for D∗ = 1.0 and St = 0.25 in Figure 12. Here, ω∗ = ωC/U∞ , where ω = ∇ × u is
the vorticity, and u ∗ = ū/U∞ , where ū is the time-averaged streamwise velocity. For this phase
difference, the coefficients of thrust and power are reduced when compared to the isolated foil
(Figure 6). The instantaneous vorticity fields for both the experiments and the vortex array model
indicate that pairs of vortices are generated that mutually induce themselves towards the centerplane
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FIG. 13. Wake structures for D∗ = 1 and φ = π . Instantaneous vorticity fields for (a) vortex array model and (c) PIV
experiments. Time-averaged velocity fields for (b) vortex array model and (d) PIV experiments. Vortex labels are the same
as in Figure 12.
between the foils. In both the model and the PIV results the vortex pairs are labeled A2 /B2 , B1 /C1 ,
and C2 /D2 .
The vortices shed into the wake remain in a reverse von Kármán vortex street alignment that is
angled towards the other foil. The width of the wake becomes compressed, and the time-averaged
momentum jets for this in-phase case converge towards one another, and in the case of the vortex
array model coalesce into a single jet downstream of the foils. The limited field of view for the PIV
prohibited the latter observation. The secondary-instability that generated a shear layer roll-up in
the hydrogen bubble visualizations (Figure 10) is also observed in the PIV results. In particular, we
observe a series of smaller secondary vortices trailing behind vortex B1 . Other qualitative differences
between the bubble visualizations and the PIV results are probably due to the differences in Strouhal
number (0.125 compared to 0.25, respectively).
For the out-of-phase case (φ = π ), shown in Figure 13, the thrust produced by both foils was
enhanced when compared to an isolated foil. In contrast to the in-phase case, the vortex pairs tend
to advect away from the centerplane, which produces time-averaged momentum jets angled away
from each other. As indicated earlier, the φ = π case is analogous to the near-ground case studied by
Quinn et al.,10 and the jet divergence observed here is consistent with their observations. The motion
of the vortex pairs away from the centerplane was also seen in the hydrogen bubble visualizations
(Figure 11), although the field of view was more limited.
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FIG. 14. Wake structures for D∗ = 1 and φ = π /2. Instantaneous vorticity fields for (a) vortex array model and (c) PIV
experiments. Time-averaged velocity fields for (b) vortex array model and (d) PIV experiments. Vortex labels are the same
as in Figure 12.
For the intermediate phase case (φ = π /2), shown in Figure 14, the thrust produced was
significantly higher for the foil that was leading in the oscillation cycle, and so we expect more
vorticity to be shed from that foil. The PIV results confirm this expectation, in that typical vortex
cores behind the leading foil contain around 40% more circulation than those behind the trailing
foil. This adjustment was included in the vortex array model. In this case, negative vorticity gathers
in the center of the wake, which yields a reverse von Kármán vortex street for the leading foil and
a classical von Kármán vortex street for the trailing foil, which then produces regions of high and
low time-averaged momentum behind the leading and trailing foils, respectively. This effect was
not observed when the vortices were all of equal strength in the vortex array model, implying that
this particular vorticity distribution cannot be attributed to the phase difference alone, and results
instead from a combination of the phase difference and the relative differences between strengths of
the wake vortices.
We now propose possible mechanisms for the development of the different wake configurations,
as illustrated in Figure 15. Consider first the case of φ = 0. According to the Biot-Savart law, the
induced velocity of vortex 1 causes vortex 1′ to lag behind and pair up with vortex 2′ . This vortex
pair then advects towards the centerplane. At the same time, the induced velocity of vortex 2′ causes
vortex 2 to lag behind and pair up with vortex 3. The cycle repeats itself, with vortex pairs forming
and advecting inward, creating two time-averaged momentum jets that converge. For the case of
φ = π , vortices 1 and 1′ both lag behind due to the mutually induced velocities. Vortices 1 and 1′
now pair up with vortices 2 and 2′ , respectively, and these pairs advect away from the centerplane
forming two diverging time-averaged momentum jets in the wake. In the case of φ = π /2, the induced
velocity of vortex 1 causes vortex 1′ to lag behind and move in the negative y-direction. Vortex 1
then causes vortex 2′ to lag behind, and to move in the positive y direction. The result is a clockwise
rotation of vortices 1, 2, 1′ , and 2′ . In the final orientation, negative vorticity is concentrated along
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2
1
1
2
3
2’
1’
2’
1’
1’
3’
2
2
1
3
1
1
1’
2’
3’
1’
2’
1’
2
2
1
1’
1
1
3
1’
3’
1’
2’
2’
FIG. 15. Proposed interpretation of the wake dynamics. Top row: φ = 0. Middle row: φ = π /2 (top foil leads). Bottom row:
φ = π.
the centerplane, with positive vorticity on either side. A time-averaged momentum jet forms behind
the leading foil, as well as a region of slow moving fluid behind the trailing foil, consistent with the
observation that the leading foil is producing more thrust.
While the current effort focuses on the wake dynamics of a pair of tandem foils in a side-byside configuration, the case of N > 2 foils is also relevant to bio-inspired propulsion and potential
energy harvesting applications. Preliminary investigations into the wake dynamics for the case with
more than two foils in a side-by-side configuration using the vortex array model indicate that the
general mechanisms described for the wake evolution here appear to extend to cases with N > 2.
Nevertheless, the precise nature of the wake dynamics and propulsive performance for more than
two foils in a side-by-side configuration is likely to depend on the specific parameters for a given
configuration and is left for future research.
VII. CONCLUDING REMARKS
The propulsive characteristics of unsteady foils oscillating in a side-by-side configuration were
found to depend on both the spacing and phase differences between the foils. Three general observations were made: (1) in-phase oscillation reduces the thrust production and power consumption
for each foil, (2) out-of-phase oscillation enhances the thrust produced and power consumed, and
(3) oscillation at intermediate phases enhances the thrust produced and power consumed by one of
the foils and reduces them for the other foil. The propulsive efficiency for each foil was found to
improve for in-phase oscillations, while it remained approximately unchanged for all other configurations. These observations held for all foil spacings considered, but the magnitudes of the benefit
(or deficit) were found to decay with foil separation according to ∼D∗−0.4 . At the smallest foil
spacing considered (D∗ = 0.5), the thrust could be enhanced by up to about 70% when the foils
were oscillated out-of-phase, and the efficiency could be enhanced by up to about 30% when the
foils were oscillated in-phase.
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These observations suggest that the various oscillation strategies that could be employed by a
propulsor will have trade-offs that depend on the objective. For propulsors or agents that are working
cooperatively, the propulsive efficiency can be enhanced at the cost of thrust production by employing
in-phase oscillations. In contrast, the thrust could be improved with no change in efficiency due to
an increased power requirement by employing out-of-phase oscillations. In the intermediate case,
one of the foils produces more thrust than the other. If both foils were attached to the same body, for
instance a bio-inspired underwater vehicle, this strategy might be well suited for maneuvering due
to the imbalance in propulsive forces without a decline in the net thrust of the system.
For agents that are not working cooperatively but are instead competing for a particular resource,
other strategies are available. In particular, it was shown that by oscillating in-phase or out-of-phase
with a neighbor that agent would gain no propulsive advantage in comparison to its neighbor. It
would be possible to gain an advantage over a neighbor by leading in the oscillation period since
thrust and efficiency are enhanced, while the other agent would experience a significant performance
reduction. This strategy could be useful for animals while feeding, evading a predator, or perhaps
impressing a potential mate. Based on these observations, it is not surprising that there is no uniform
strategy employed in nature.18 Swimming and flying animals must adapt to a variety of conditions
and situations, and it is likely to be beneficial to them to be able to continuously alter their oscillation
strategies in order to achieve a specific objective.
Flowfield analysis indicated that the wakes shed by foils are complex and depend critically
on the phase differential between the foils. Flow visualizations showed that for a given Strouhal
number, different instabilities can be excited depending on the phase difference. In particular, a
secondary instability that leads to a Kelvin-Helmholtz like shear layer roll-up was present as the
foils oscillated in-phase. This instability was suppressed as the foils were oscillated out-of-phase,
and the only vortices observed in the wake were generated as the foils reversed their direction. We
also observed three distinct wake structures. For in-phase oscillations, vortex pairs form in the wake
of each foil that move by mutual induction towards the centerplane between the foils. In contrast,
for out-of-phase oscillations, vortex pairs form in the wake of each foil that induce themselves
away from the centerplane between the foils. For the intermediate case (φ = π /2), the vortices
produced by the foils are of different strengths. The foil that is leading in the oscillation period not
only produces more thrust, but also generates vortices that contain more circulation in comparison
to the other foil. The vortex interactions in this instance yield a reverse von Kármán vortex street
in the wake of the foil leading in the oscillation period and a regular von Kármán like vortex street
in the wake of the other foil. Despite the rich complexity of the flowfields in the wake of the foils,
the qualitative vortex dynamics are shown to be well described by inviscid considerations captured
by a vortex array model.
ACKNOWLEDGMENTS
The authors would like to thank Jessica Shang for aiding in the implementation of the hydrogen
bubble visualization experiments. The work was performed with support provided by the Office of
Naval Research under Program Director Dr. Bob Brizzolara, MURI Grant No. N00014-08-1-0642.
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