pnasresearcharticle Kurt Fish schools are fascinating examples of self-
organization in nature. They serve many purposes from enhanced foraging, and
protection to improved socialization and migration. However, our understanding
of the hydrodynamic interactions in schools is primitive. It has been
postulated that these interactions can regulate energy usage and speed, as
well as push and pull individuals thereby altering the school’s structure and
function. We have discovered that stable arrangements of swimmers can arise in
two-dimensional schools through passive hydrodynamic forces alone. In these
stable arrangements, swimmers also experience speed and efficiency benefits.
This opens the door to considering that the structure and function of fish
schools may be more strongly regulated by hydrodynamic interactions than
previously known. M. K. helped design the study, gathered and processed
constrained-foil measurements, and drafted the manuscript. P. C. O. helped
design the study, gathered and processed the unconstrained-foil data, and
helped revise the manuscript. A. M. helped design the study, gathered and
processed the numerical and the flow-visualization data, and helped revise the
manuscript. K. W. M. helped design the study, and helped revise the
manuscript. The authors declare no conflict of interest. 1To whom
correspondence should be addressed. E-mail<EMAIL_ADDRESS>
# Two-dimensionally stable self-organization arises in simple schooling
swimmers through hydrodynamic interactions

Melike Kurt  Aerodynamics and Flight Mechanics Group, Faculty of Engineering
and the Environment, University of Southampton, Southampton SO17 1BJ, UK
Pedro Costa Ormonde  Mechanical Engineering and Mechanics, Lehigh University,
Bethlehem, PA, 18015, USA  Amin Mivehchi  Mechanical Engineering and
Mechanics, Lehigh University, Bethlehem, PA, 18015, USA  Keith W. Moored
Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA, 18015,
USA

###### Abstract

We present new constrained and free-swimming experiments and simulations of a
pair of pitching hydrofoils interacting in a minima school. The hydrofoils
have an out-of-phase synchronization and they are varied through in-line,
staggered, and side-by-side arrangements within the two-dimensional
interaction plane. It is discovered that there is a two-dimensionally stable
equilibrium point for a side-by-side arrangement. In fact, this arrangement is
super-stable, meaning that hydrodynamic forces will passively maintain this
arrangement even under external perturbations and the school as a whole has no
net forces acting on it, causing it to drift to one side or the other.
Moreover, previously discovered one-dimensionally stable equilibria driven by
wake vortex interactions are shown to be, in fact, two-dimensionally unstable,
at least for an out-of-phase synchronization. Additionally, it is discovered
that a trailing-edge vortex mechanism provides the restorative force to
stabilize a side-by-side arrangement and the stable equilibrium is further
verified for freely-swimming foils where dynamic recoil motions are present.
When constrained, the swimmers experience a collective thrust and efficiency
increase up to 100% and 40%, respectively, in a side-by-side arrangement,
whereas the staggered arrangements output an even higher efficiency
improvement of 87% with a 94% increase in thrust. For freely-swimming foils,
the recoil motion attenuates the improvements at the stable equilibrium,
showing a more modest speed and efficiency enhancement of up to 9% and 6%,
respectively. These newfound schooling performance and stability
characteristics suggest that fluid-mediated equilibria may play a role in the
control strategies of schooling fish and fish-inspired robots.

###### keywords:

collective locomotion $|$ hydrodynamic interactions $|$ fish schooling $|$
pattern formation $|$ collective performance

Self-organization of living systems is one of Nature’s most ubiquitous and
mesmerizing phenomena. It arises across a wide range of spatial and temporal
scales from the cells in our bodies (1) and swarming of microorganisms (2) to
the flocking of birds (3) and schooling of fish (4). For macroscopic flyers
and swimmers, a wide range of hypotheses have attributed collective behavior
to social interactions (5), protection against predators (6), food prospect
optimization (7), and/or energetic benefits (4). Our knowledge of the latter
hypothesis is limited since it is regulated by complex hydrodynamic
interactions. Yet, both the spatial organization (3, 8) and temporal
synchronization (9, 10, 11) have emerged as factors influencing the
hydrodynamic interactions, and, consequently, the energetic cost of locomotion
and traveling speed of individuals in a collective. Still, our understanding
of the force production and energetics of schooling swimmers is mostly limited
to canonical spatial arrangements such as a leader-follower in-line
arrangement (12, 13, 14, 15) and a side-by-side arrangement (16, 17, 11, 18),
while there are fewer studies of staggered arrangements (19, 20, 21, 22).

Because of these studies it is commonly presumed that the spatial organization
observed in schools is driven by the interest to maximize swimming efficiency
or force production. However, another explanation was first proposed by Sir
James Lighthill (23). The so-called Lighthill conjecture (24) postulates that
the arrangements of fish in a school may be due to the interaction forces that
push and pull the swimmers into a particular stable formation, much like the
atoms in a crystal lattice. Indeed, this idea of passive self-organization has
shown promise in recent studies where one-dimensional streamwise stability has
been observed in pairs of in-line self-propelled foils (25, 24) or in small
schools of various arrangements (20, 26), as well as in pairs of in-line
hydrofoils with differing kinematics (27). While these studies have shown
seminal results supporting the Lighthill conjecture, they have only probed the
one-dimensional stability of arrangements. However, two-dimensionally or even
three-dimensionally stable arrangements are required for the passive self-
organization of schools that produce two-dimensional or three-dimensional
flows.

Here, we advance our understanding of the hydrodynamic interactions of
schooling swimmers in two ways. For the first time, we measure the two-
dimensional stability of schooling arrangements for constrained and
unconstrained foils, which takes us closer to understanding the role of the
Lighthill conjecture in schooling formations. We discover that many of the
one-dimensionally stable formations previously observed are, in fact, unstable
once the cross-stream stability is considered. Yet, we still find that a side-
by-side arrangement is two-dimensionally stable providing support for the
hypothesis that this arrangement observed in real fish (11) may be due to
passive self-organization. Second, we measure the force production and
energetics of two interacting hydrofoils throughout a plane of possible
arrangements ranging from in-line to side-by-side by passing through the
possible staggered arrangements. We reveal that there is a thrust and
efficiency optimum in a slightly-staggered arrangement where there is direct
vortex impingement on the follower.

## Experimental Approach and Results

To examine the flow interactions that occur in schools, full swimmer models
can be readily used in numerical studies (19), however, these models are
difficult to implement experimentally. Instead, experiments typically use
oscillating hydrofoils as a simple model of the propulsive appendages of
animals (12, 16, 17, 25, 24, 13, 14, 27, 28, 22). Importantly, these
oscillating hydrofoils capture the salient unsteady fluid mechanics of the
added mass forces, circulatory forces, and shed vortices.

Figure 1: Schematic of positions of the follower hydrofoil relative to the
leader. Shown are two coarse rectangular grids of $0.1$c spacing ranging from
$-0.2\leq X^{*}\leq 2$, $0.5\leq Y^{*}\leq 0.8$, and $1.1\leq X^{*}\leq 2$,
$-0.2\leq Y^{*}\leq 0.4$, as well as a refined rectangular grid of $0.05$c
spacing ranging from $1.05\leq X^{*}\leq 1.15$, $-0.15\leq Y^{*}\leq 0.15$. In
total there are 180 grid points.

Following this simple model approach, experiments were conducted on a pair of
pitching hydrofoils immersed in a water channel, forming a minimal school or
minimal collective. The flow over the hydrofoils was restricted to be
nominally two-dimensional. The arrangement of the hydrofoils was varied by
manipulating the follower hydrofoil position in the streamwise and cross-
stream directions as shown in Figure 1. The dimensionless distances were
normalized by the chord length as $X^{*}=x/c$, and $Y^{*}=y/c$. The leader
hydrofoil was prescribed a sinusoidal pitching motion of
${\theta}_{L}(t)={\theta}_{0}\sin(2\pi ft)$, where the oscillation frequency
is $f$, and the pitching amplitude is $\theta_{0}$, which is related to the
peak-to-peak trailing edge amplitude as $A=2c\sin(\theta_{0})$. The follower
was pitched similarly as ${\theta}_{F}(t)={\theta}_{0}\sin(2\pi ft+\phi)$ with
a fixed phase difference or synchrony of $\phi=\pi$ throughout the study.
Moreover, the oscillation frequency and the dimensionless amplitude,
$A^{*}=A/c$, were also fixed throughout the study at $f=0.98$ Hz and
$A^{*}=0.25$, which gives a fixed reduced frequency of $k=fc/U=1$, and a fixed
Strouhal number of $St=fA/U=0.25$. These dimensionless numbers are typical of
efficient biological swimming (29, 30). Direct force measurements were taken
from each hydrofoil at every ($X^{*}$, $Y^{*}$) position, which corresponds to
a total of 180 arrangements in the $x$-$y$ plane. For further information
about the water channel setup, actuator mechanisms, sensors, the methods used,
and the definition of the performance coefficients for an individual hydrofoil
as well as the collective, please see Materials and Methods and Supplementary
Information.

### Follower Force Map

In order to probe the Lighthill conjecture in two-dimensions, the relative
forces acting on the follower in the ($x$-$y$) interaction plane must be
examined. This is done by constructing a force map, which is described below.

Consider a frame of reference attached to the leader as in Figure 1. The
relative lift, $\Delta L$, in the cross-stream direction and the relative
thrust, $\Delta T$, in the streamwise direction are defined simply as a
difference between the forces acting on the two hydrofoils as $\Delta
T=T_{F}-T_{L}$ and $\Delta L=L_{F}-L_{L}$, where forces acting on the leader
and follower hydrofoils are denoted with $(.)_{L}$ and $(.)_{F}$,
respectively. Figure 2A and 2B show the relative force conditions that leads
to the follower either moving towards or moving away from the leader in the
streamwise ($x$) and cross-stream ($y$) directions.

Figure 2: Typical conditions leading to positive and negative relative (A)
lift and (B) thrust. (C) Follower force map with an out-of-phase synchrony
between the leader and follower, i.e. $\phi=\pi$. The arrows on the force
lines indicate the direction that the follower would move in relative to the
leader if it were free-swimming. The green and red circles represent the
stable and unstable equilibria, respectively.

First, consider the relative lift for the positive $x$-$y$ plane. The follower
is pushed away from the leader in the cross-stream direction ($\updownarrow$)
when the relative lift force is greater than zero (Figure 2A). This condition
arises either when lift forces acting on the foils are in the same direction
and $L_{F}>L_{L}$, or when they are acting in opposite directions and pointing
away from each other ($L_{L}<0\downarrow$, $L_{F}>0\uparrow$). In contrast,
the follower is pulled towards the leader when the lift forces are acting in
the same direction and $L_{L}>L_{F}$ or acting in opposite directions and
pointing towards each other ($L_{L}>0\uparrow$, $L_{F}<0\downarrow$).

Next, consider the relative thrust force in the positive $x$-$y$ plane. A
positive relative thrust force ($\Delta T>0$) acts to move the follower
towards the leader, which arises when $T_{F}>T_{L}$. In contrast, when
$T_{L}>T_{F}$ the relative thrust force is negative ($\Delta T<0$) and the
follower moves away from the leader in the streamwise direction as shown in
Figure 2B. If $T_{L}=T_{F}$ then the leader and follower swim at the same
speed and do not move closer or apart.

To visualize the directions of the relative forces acting on the follower
throughout the $x$-$y$ plane we constructed a force map, which is a novel
visualization made up of force lines (Figure 2C). Put simply, the force map
conveys the direction that the follower would move in as observed by the
leader. The force map is constructed with the origin located at the leading
edge of the leader and a relative force vector
($\mathbf{F}_{\text{rel}}=-\Delta T\mathbf{\hat{x}}+\Delta L\mathbf{\hat{y}}$)
is determined at each of the measurement positions detailed in Figure 1, such
that a relative force vector field is created. Force lines are then graphed as
lines that are everywhere tangent to the local relative force vector field,
analogous to streamlines.

### Observed Equilibria

The force map reveals four critical points or equilibria where the relative
force vector is equal to zero, that is, $\Delta T=0$ and $\Delta L=0$, which
are marked by green and red circles in Figure 2C. The first equilibrium point
is located at $(X^{*},Y^{*})=(0,0.6)$ in Region 1 where the leader and
follower are interacting in a side-by-side arrangement. Interestingly, as the
force lines merge at this point, their direction indicates that this critical
point is a stable equilibrium point in two-dimensions. Therefore, when any
perturbations move the follower away from this point, forces will arise to
return the foil back to this location. Previous low Reynolds number
simulations have also shown that side-by-side arrangements of bending foils
are two-dimensionally stable (20), at least for foils constrained in the
cross-stream direction. Another critical point in Region 2 is located at
$(X^{*},Y^{*})=(1.2,0)$ in the leader’s wake where the follower is directly
in-line with the leader. Region 2 represents an equilibrium point that is
stable to streamwise perturbations, but unstable to cross-stream
perturbations, that is, an unstable saddle point. In Region 3 there is an
equilibrium point located at $(X^{*},Y^{*})=(1.6,0.3)$ representing a
staggered arrangement of the foils. This equilibrium point represents an
unstable source point where the force-lines spiral outward from it. Finally,
Region 4 reveals another unstable saddle point for a staggered arrangement
with the follower located at $(X^{*},Y^{*})=(1.5,0.7)$.

By definition the mean forces go to zero at the equilibria, however, the
thrust and lift forces exhibit fluctuations as the foils undergo pitching
motions. In order to understand this aspect, we analyzed the lift and thrust
fluctuations of the leader and follower as presented in Figure S3 in the
Supplementary Information. Overall, the lift fluctuations are found to be an
order of magnitude higher than the thrust fluctuations for the constrained
foils, as previously observed in (31) for a pair of interacting fish through
3D computations. At the stable side-by-side equilibrium, the lift and thrust
fluctuations increase by 30-40% over an isolated hydrofoil. When unconstrained
(freely swimming), this may cause the foils to undergo larger amplitude recoil
motions and the arrangement to no longer be stable. Around the in-line
unstable equilibrium, the force fluctuations on both foils are, surprisingly,
reduced compared to an isolated hydrofoil. At the other equilibria, the
fluctuations stay effectively at about the same levels as an isolated
hydrofoil. Also, in the wake of the leader around $1.5\leq X^{*}\leq 2$, there
is an increase in the follower’s lift and thrust fluctuations by 20-40%.

### Collective Performance

Beyond probing the Lighthill conjecture in two dimensions, the collective
performance was also measured in the same interaction plane. Figure 3 presents
the normalized collective thrust, $C^{*}_{T,C}$, and power, $C^{*}_{P,C}$, the
collective lift, $C_{L,C}$, and the normalized collective efficiency,
$\eta_{C}^{*}$, as a contour map of $(X^{*}$, $Y^{*})$. The normalized
performance metrics compare the collective performance of the leader-follower
pair with that of two isolated hydrofoils. Note that in contrast to the
relative forces that were discussed in the previous section, a collective lift
of zero does not necessarily mean zero lift for the foils individually.

Figure 3: Contour maps of normalized collective (A) thrust, and (B) power, (C)
collective lift, and (D) normalized collective efficiency. The black dashed
lines show the locations corresponding to $X^{*}=0$, $Y^{*}=0$, and
$Y^{*}=0.125$. The leader position, size, and amplitude of motion is shown for
reference. The green and red circles represent the stable and unstable
equilibria, respectively, from Figure 2.

Figure 3A shows the normalized collective thrust as a function of the
streamwise and cross-stream position of the follower relative to the leader.
Generally, the collective thrust stays within the range of $1\leq
C_{T,C}^{*}\leq 2$, which indicates that the collective performs better than
two hydrofoils in isolation for nearly the entire interaction plane considered
here. Previous work has also observed similar thrust enhancements albeit for a
limited set of data of one side-by-side and two staggered arrangements at
$Y^{*}=1$ (32). The peaks in collective thrust can be grouped into two
regions. First, is a region enclosing the side-by-side arrangements, along the
line of $X^{*}=0$ (black dashed line), where the collective is found to
achieve $40-100\%$ higher thrust than in isolation. Recent work has
established (33) that the thrust increase for out-of-phase pitching foils in
this region originates from an increase in their added mass. It was determined
that the added mass thrust dominates the thrust production of pitching foils
and thereby wake effects play no significant role in their thrust increase.
However, for heaving or combined heaving and pitching foils, wake effects, and
circulatory forces in general, are expected to be important for thrust
production in a side-by-side arrangement (34). A second region corresponds to
the direct wake interactions enclosing the in-line and staggered arrangements,
within the region of $0\leq Y^{*}\leq 0.3$. For the in-line arrangements, the
collective thrust reaches a $74\%$ peak increase over the hydrofoils in
isolation, whereas, the thrust reaches a $94\%$ peak increase for slightly-
staggered arrangements along the $Y^{*}=0.125$ line. This line is where direct
wake vortex impingement onto the follower is anticipated. It has already been
established that wake-body impingement interactions generate increased thrust
through an increase in the effective angle of attack of the follower and are
therefore circulatory in nature (12, 13).

Figure 3B presents the normalized collective power in the interaction plane.
For near wake interactions where the collective is in in-line or staggered
arrangements ($X^{*}>1.1$ and any $Y^{*}$), the collective power exhibits
little variation from the isolated case, whereas, the side-by-side
arrangements result in up to a 50% increase in power.

Similarly, the collective lift coefficient is presented in Figure 3C. For
side-by-side arrangements, where a stable equilibrium point is located, the
collective lift generation is found to be negligible. This means that when two
individuals are swimming in this stable arrangement, the arrangement is, in
fact, super-stable, that is, the relative distances between the swimmers do
not change and the pair of swimmers will remain swimming forward without a
collective drift to one side or another. Likewise, for the in-line interaction
region, where an unstable equilibrium point is located, collective lift
generation is found to be negligible as well.

In Figure 3D, the collective efficiency is observed to increase by $10-40\%$
over that of isolated foils for the side-by-side interaction region, which has
been observed previously (32). In the in-line interaction region, even higher
peak efficiency increases are identified with up to a $74\%$ increase for in-
line interactions and an $87\%$ increase for slightly-staggered arrangements
along the $Y^{*}=0.125$ dashed line. It is clear that while both the side-by-
side and in-line interaction regions see comparable thrust increases, the
efficiency increase in the side-by-side interaction is tempered by a
concurrent rise in power, whereas the efficiency increase in the in-line
interaction region is solely driven by the increase in thrust. Some previous
work has not observed an efficiency benefit for staggered arrangements with
$Y^{*}=1$ (32), however, in the near-wake interactions probed in the current
study with $Y^{*}\leq 0.8$ it is revealed that the greatest efficiency
benefits are observed for slightly staggered arrangements.

### Wake Dynamics and Time-Varying Forces

Figure 4: (A) Vorticity field for the stable side-by-side arrangement
($X^{*}=0$, $Y^{*}=0.6$) at $t^{*}=0.5$ when the foils are pitching away from
each other. (B) Vorticity field for the arrangement of $X^{*}=0.1$,
$Y^{*}=0.6$ at $t^{*}=0.5$. (C) The time-varying thrust coefficient of the
leader and follower at $X^{*}=0.4$, $Y^{*}=0.6$. (D) Schematic of proposed
trailing-edge vortex mechanism responsible for a restorative force back to the
stable side-by-side arrangement.

To understand the flow mechanisms behind the stable side-by-side arrangement
and the high-efficiency slightly-staggered arrangement particle image
velocimetry (PIV) measurements are paired with the leader and follower’s time-
varying forces.

Figure 5: Vorticity field at $t^{*}=1$ during the leader’s up-stroke and the
follower’s down-stroke for (A) the in-line arrangement of $X^{*}=1.7$ and
$Y^{*}=0.0$, (B) the staggered arrangement of $X^{*}=1.7$ and $Y^{*}=0.125$,
and (C) the staggered arrangement of $X^{*}=1.7$ and $Y^{*}=0.25$. Time-
varying thrust coefficient of (D) the leader and (E) the follower, for
arrangements of $X^{*}=1.7$ and $Y^{*}=0.0,\,0.1,\,0.2,\,0.3$. (F) The cycle
average $x$-component of velocity, $u/U_{\infty}$ at the midpoint between the
trailing edge of the leader and the leading edge of the follower at
$X^{*}=1.35$ for three arrangements of $X^{*}=1.7$ and $Y^{*}=0.0,\,0.125$ and
$0.25$. $u/U_{\infty}$ presented in (F) is calculated based on the time-
averaged flow-fields for the arrangements, (G) $X^{*}=1.7$ and $Y^{*}=0.0$,
(H) $X^{*}=1.7$ and $Y^{*}=0.125$, and (I) $X^{*}=1.7$ and $Y^{*}=0.25$.

Figure 4A presents the vorticity shed from the leader and follower at the
stable side-by-side equilibrium at a dimensionless time of $t^{*}=t/T=0.5$
where the foils are pitching away from each other. The forming clockwise
vortex of the follower and counterclockwise vortex of the leader mutually
induce each other in a way that slows their downstream advection. These
vortices then pair with their counter-rotating counterparts shed a half cycle
later. The pairs mutually induce away from the symmetry line between the
foils, leading to momentum jets behind the foils that deflect away from the
symmetry line. The deflected jet and vortex pairing mechanism were first
observed in (17) and later in subsequent work (35).

The cross-stream stability of the side-by-side arrangement is mediated by a
balance of the wake-induced forces from the deflected jet and the quasi-steady
forces, that is, the body-body flow interaction between the foils, which has
been established in (28). Furthermore, studies examining the interaction of
two side-by-side undulatory swimmers have also observed deflected wake
interactions albeit with a less pronounced deflection (36). In fact, for some
cases of undulatory swimmers very little if any jet deflection is noticeable
(37, 38) indicating that the lift forces mediating the stable equilibria may
be quite weak in some cases as observed in (37).

It has already been established in (33) that the thrust increase with
decreasing separation distance between the side-by-side leader and follower
foils originates from an increase in the added mass thrust of the two foils,
which dominates the thrust production of pitching foils (39, 40). The increase
in added mass and influence of the shedding vortices from the interacting
foils also increase the power, though, not quite as much as the thrust giving
rise to a small increase in efficiency as the foils get closer to each other
(33).

What is not established yet is the mechanism that generates restorative thrust
forces for the leader and follower to bring them back to the side-by-side
equilibrium when one of them is perturbed downstream of the other. Figure 4B
shows the vorticity field at a dimensionless time of $t^{*}=0.5$ when the
follower is located slightly downstream of the side-by-side equilibrium
arrangement at $(X^{*},\,Y^{*})=(0.1,\,0.6)$. The wakes of the leader and
follower show the same vortex pairs as observed at the equilibrium arrangement
with a slight asymmetry. Figure 4C presents the time-varying thrust of the
leader and follower for an arrangement that is perturbed further downstream at
$X^{*}=0.4$ and $Y^{*}=0.6$ where the difference in the leader and follower
thrust is more pronounced. When the follower is perturbed downstream it
produces more thrust than the leader around $t^{*}=0.5$, which acts to
increase the swimming speed of the follower and restore the two foils back to
the stable equilibrium. Figure 4D shows a schematic that describes the flow
mechanism behind these restorative forces. When the follower is perturbed
downstream, the forming counterclockwise vortex of the leader is closer to the
leading edge of the follower (as observed in 4B), which leads to an increase
in the downwash acting on the follower. Since the foils are pitching away from
each other, this downwash increases the effective angle of attack of the
follower, thereby increasing its thrust production. In the same arrangement,
the forming clockwise vortex from the follower is farther from the leading
edge of the leader than at the equilibrium arrangement. This reduces the
upwash acting on the leader, leaving a stronger downwash from its own forming
vortex. This downwash acts to decrease the effective angle of attack of the
leader thereby decreasing its thrust production. This trailing-edge vortex
mechanism also applies in a similar way to the case where the follower is
perturbed upstream of the equilibrium arrangement.

Figure 5A, 5B, and 5C present flow measurements around the high collective
efficiency arrangement where $X^{*}=1.7$ and $Y^{*}=0$, 0.125, 0.25,
respectively, at the dimensionless time of $t^{*}=1$. It is observed that the
countercloskwise red vortex shed from the leader directly impinges on the
follower at $Y^{*}=0.125$. Around $t^{*}=1$ the time-varying thrust of the
follower presented in Figure 5D is seen to maximize at $Y^{*}=0.1$ with a
pronounced reduction in the thrust for $Y^{*}>0.2$. This can be linked to the
impinging vortex, which, at this time, induces an upwash on the follower while
the follower is pitching through its down-stroke. The upwash then acts to
increase the effective angle of attack and thereby increase the thrust
production. For the direct impingement location of $Y^{*}=0.125$ this effect
is maximized, which helps to drive the high efficiency of the collective.
However, the leader’s thrust also plays a role since it slightly increases as
the follower moves away from the in-line arrangement to a slightly-staggered
arrangement. Figure 5E shows that around $t^{*}=1$ the time-varying thrust of
the leader is increased slightly when $Y^{*}\geq 0.1$. This can be understood
by examining the mean velocity in the $x$-direction, $u$, behind the leader
shown in Figure 5G, 5H, and 5I. The momentum jet behind the leader is observed
to slightly increase its strength as the follower moves away from the in-line
arrangement to a slightly-staggered arrangement. This can be quantified by
extracting mean velocity profiles at $X^{*}=1.35$, halfway between the leader
and follower, and calculating their momentum flux. The extracted profiles are
plotted in Figure 5F, and they are used to calculate the momentum flux
coefficient as,

$C_{p_{x}}\equiv\frac{\overline{p}_{x}}{1/2\,\rho
U_{\infty}^{2}cs}=\int_{Y^{*}}\left(\frac{u}{U_{\infty}}\right)^{2}\,dY^{*}.$
(1)

The momentum flux coefficients for the $Y^{*}=0$, 0.125, and 0.25 cases are
$C_{p_{x}}=2.74$, 2.75, and 2.76, respectively. This shows a slight increase
concurrent with the increase in thrust of the leader as the follower moves
from an in-line arrangement to a slightly-staggered arrangement. Taken
together the strengthening of the leader’s jet and the direct vortex
impingement on the follower lead to a maximum in the collective thrust and
efficiency for a slightly-staggered arrangement at $X^{*}=1.7$ and
$Y^{*}=0.125$.

### Dynamic Schooling Interactions

Within the interaction plane only one two-dimensionally stable equilibrium
point has been discovered for a side-by-side arrangement. However, this result
neglects the effect of dynamic recoil motion of freely swimming bodies, which
may alter the physics of these interactions. Dynamic recoil motion occurs when
a two-dimensionally unconstrained pitching hydrofoil, for example, pitches
through its downstroke. During this stroke a positive lift force is generated
leading to a recoil motion where the body will heave upward in response to the
lift. In this sense, the recoil motion introduces heaving that lags the
pitching motion by nearly 180o, which is known to lower the thrust production
(41) and may alter the lift and the stability of an equilibria. In order to
explore the effect of these dynamic recoil motions on the stability of the
side-by-side equilibrium point, to further verify the findings from the force
map, and to determine the free-swimming collective performance, freely
swimming experiments and simulations were performed.

Figure 6: (A) Equilibrium distance of the in-line arrangement for streamwise
unconstrained and cross-stream constrained experiments. The distance $S$ is a
function of the synchrony phase $\phi$. Marker color is mapped to the initial
schooling number. (B) Time-varying trajectories of a fully unconstrained
follower (free to move in the $X^{*}$-$Y^{*}$ plane) starting from six
different in-line arrangements. The markers represent the position of the
follower’s leading edge and their color is mapped to the dimensionless time
$t/T$. Dashed lines represent the domain boundaries.

In the freely swimming experiments the leader foil is still constrained to a
fixed position, however, the follower foil is free to move in the
$X^{*}$-$Y^{*}$ plane. This is achieved by mounting the follower foil
actuation mechanism, for the first time, to a novel double air bearing stage
for nearly friction-less motion, and with on-board batteries and wireless
communication to eliminate forces from electronic wiring. Exceptional care is
taken to align, level, and counter-bend the air-bearing rails in order to
minimize these sources of non-hydrodynamic forces acting on the foil. For more
details on the unconstrained experimental approach, please see Unconstrained
Foil Experiments.

To verify that these novel experiments are measuring the actual hydrodynamic
forces acting on interacting foils instead of being corrupted by non-
hydrodynamic forces, such as settling to a false equilibrium due to rail
bending, we have reproduced results of flow-mediated streamwise stable
equilibria discovered in (27). Figure 6A presents data from experiments where
the follower foil is downstream of the leader in an in-line arrangement.
Importantly, the follower is constrained in the cross-stream direction, but
unconstrained in the streamwise direction. The graph shows the final
dimensionless gap distance of the follower and its final dimensionless
streamwise spacing on the vertical left and right axes, respectively, as a
function of the synchrony of the follower. The gap distance, $g=X^{*}c-1$, is
defined in (27) as the distance from the trailing edge of the leader to the
leading edge of the follower, and the vortex wake wavelength, $\lambda$
(measured from particle image velocimetry measurements of an isolated pitching
foil), define the dimensionless gap distance or so-called schooling number
(25). The initial schooling number defined by the initial distance between the
leader and the follower is denoted from small to large values by markers
colored from black to white. Note that for these measurements the entire range
of synchrony ($0-2\pi$) was examined.

Depending upon the initial schooling number and the synchrony, the follower
settles to a streamwise stable equilibria in $\mathcal{O}(10)$ cycles. The
final schooling number follows a linear relationship with the synchrony, and
for a specific synchrony such as $\phi=\pi$, there are multiple final
locations for the follower depending upon its initial schooling number. In
fact, the final schooling numbers for $\phi=\pi$ within the data range are
$S=1.5$ and 2.5, which are precisely the same schooling numbers found in (27)
for out-of-phase heaving foils. These results validate the capability of the
experiments to measure flow-mediated equilibria instead of experimental
artifacts.

In a second experiment the follower starts at a range of in-line arrangements,
but it is unconstrained in both the streamwise and cross-stream directions.
This experiment examines the two-dimensional stability of in-line
arrangements, which are predicted by the force map data to be unstable saddle
points. Figure 6B presents the time-varying trajectories in the
$X^{*}$-$Y^{*}$ plane of the follower starting from various initial
arrangements. Each datum marks the cycle-averaged position and is colored by
the dimensionless time. Depending upon the initial starting position, the
follower will either move towards or away from the leader indicating that
there is a streamwise unstable equilibria between the starting position
clusters. However, the follower gets diverted in the cross-stream direction
from the $Y^{*}=0$ position and does not settle to a two-dimensionally stable
equilibria within the domain boundary as predicted by the force map data. This
verifies the finding that equilibria in the wake of the leader are indeed two-
dimensionally unstable even up to $X^{*}=4.5$.

Figure 7: Equilibrium position of the side-by-side arrangement for a fully
unconstrained follower free to move in the $X^{*}$-$Y^{*}$ plane. The
equilibrium position markers represent the final positions of individual
experiments and their color is mapped to the synchrony of the follower in the
range of $170^{\circ}\leq\phi\leq 190^{\circ}$. An average equilibrium
position is found at
$(\overline{X}^{*}_{eq},\overline{Y}^{*}_{eq})=(-0.08,0.92)$. The resulting
formation achieves a normalized swimming speed of $u^{*}=1.18$, which is 18%
higher than that of an isolated swimmer.

In a third experiment the domain boundary is shifted such that a side-by-side
arrangement may be achieved and the follower is unconstrained in both the
streamwise and cross-stream directions while the leader remains fully
constrained. From the previous force data the follower is expected to generate
more thrust when it reaches a side-by-side arrangement and will consequently
swim faster than the constrained leader if the channel flow speed remains at
the free-swimming speed for an isolated foil of $u^{iso}=0.0863\pm 0.002$ m/s.
To account for this the free-stream speed of the channel is increased until
the follower remains at a constant $X^{*}$ location without swimming upstream
of the leader. Under these conditions, Figure 7 presents the equilibrium
position of the follower relative to the leader in the $X^{*}$-$Y^{*}$ plane
for a narrow range of synchrony of $170^{o}\leq\phi\leq 190^{o}$. As predicted
by the force map for constrained foils, the unconstrained follower settles to
an equilibrium position that is nearly a perfect side-by-side arrangement with
$\left(\overline{X}^{*}_{\text{eq}},\,\overline{Y}^{*}_{\text{eq}}\right)=\left(-0.08,\,0.92\right)$.
It is also observed that this stable arrangement is robust to small variations
in the synchrony with values below and above $\phi=180^{o}$ moving the
equilibrium position slightly rearward and forward, respectively. Moreover,
for the follower to remain at a streamwise position, the free-stream speed is
increased by 18.2% to give a normalized swimming speed of the follower of
$u^{*}=1.18$. This experiment shows that there is a speed benefit in the side-
by-side arrangement and it is indeed stable even when there is dynamic recoil
motion.

Finally, to examine whether the side-by-side arrangement remains two-
dimensionally stable when both the leader and follower are free to move in the
streamwise and cross-stream directions, two freely-swimming hydrofoils are
simulated using a potential-flow-based numerical approach. For more details on
the numerical approach, please see Numerical Methods. Like the constrained
experiments, the simulated foils were prescribed sinusoidal pitching motions
about their leading edge with an out-of-phase synchrony ($\phi=\pi$). Each
foil is assigned with a mass normalized by their characteristic added mass,
$m^{*}=m/(\rho sc^{2})=2.68$, which matches previous work (28) and is
comparable to biology (for example, $m^{*}=3.86$ was calculated for cod in
(42)). Note that the constrained foil measurements presented earlier have an
effective dimensionless mass of $m^{*}=\infty$, since the foils do not exhibit
recoil motions. In the simulations, the hydrofoils have a virtual body that is
not in the computational domain, but defines parameters applying a drag force
to the hydrofoils, following a $U^{2}$ high Reynolds number drag law (39).

In free-swimming, the Strouhal number and reduced frequency are dependent
variables and are outputs of the simulations, while the Lighthill number (see
Numerical Methods) and dimensionless amplitude are input variables. By fixing
the $Li=0.3$ and varying the dimensionless amplitude over the range $0.3\leq
A^{*}\leq 0.5$, the Strouhal number is nearly constant at $St\approx 0.3$
since $U$ scales with the amplitude and the pitching frequency as $U\propto
fA$ (43, 44, 39) and the reduced frequency varies over the range $0.97\geq
k\geq 0.62$. Table 1 provides further details of the input/output variables
for the simulations.

Four cases with varying dimensionless amplitude are considered. Each case was
simulated with a range of initial positions for the follower within the region
of $-0.2\leq X^{*}_{0}\leq 0.2$ and $1.0\leq Y^{*}_{0}\leq 1.6$, with $0.1$
chord increments. An example trajectory from case IV is shown in Figure 8A to
converge to an equilibrium arrangement after 60 cycles. For each case and all
initial positions, the freely-swimming foils are found to converge to an
equilibrium point in a side-by-side arrangement as predicted by the
constrained and unconstrained foil experiments (see Supplementary Material for
all of the simulation data). This reveals that a side-by-side arrangement is
indeed a two-dimensionally stable arrangement even for freely-swimming foils
with dynamic recoil motions.

The simulation data is summarized in Table 1. Note that during self-propelled
swimming the dynamic pressure based normalization of the thrust coefficient is
equal to the Lighthill number, a fixed quantity in our simulations, so the
thrust and power coefficients in the table are normalized by the added mass
forces instead (see Numerical Methods). The data show that in each equilibrium
arrangement, the cross-stream foil spacing increases as the reduced frequency
decreases. This is the same trend observed previously for a pitching foil in
ground effect (28). The swimming speed and the thrust generation are increased
by $4-8\%$ and $9-17\%$, respectively, compared to a foil in isolation with
more compact arrangements leading to the increased gains. These performance
increases originate from increases in the added mass of the foils (see
Collective Performance) as established recently (33). As in the constrained
data, the power and efficiency also increase in a side-by-side arrangement.
The power increases by $6-11\%$, and the efficiency increases by $3-5\%$ over
a foil in isolation. The thrust and efficiency benefits are observed to be
tempered due to the dynamic recoil motions of both foils.

Table 1: Simulation input and output data for four cases of varying amplitude. There are 35 initial positions $(X_{0}^{*}$, $Y_{0}^{*})$ within the region of $-0.2\leq X^{*}_{0}\leq 0.2$ and $1.0\leq Y^{*}_{0}\leq 1.6$ with $0.1$ chord increments simulated for each case. This gives a total of 140 simulations. The equilibrium position is denoted as $(X^{*}_{\text{eq}}$, $Y^{*}_{\text{eq}})$ and the origin is defined at the leading edge of the leader. Parameters | Case I | Case II | Case III | Case IV
---|---|---|---|---
$A^{*}$ | 0.3 | 0.35 | 0.4 | 0.5
$St$ | 0.2914 | 0.2956 | 0.2997 | 0.3076
$k$ | 0.9712 | 0.8445 | 0.7493 | 0.6151
$(X^{*}_{\text{eq}}$, $Y^{*}_{\text{eq}})$ | (0, 0.7468) | (0, 0.8325) | (0, 0.9239) | (0, 1.0630)
$u^{*}$ | 1.0879 | 1.0777 | 1.0691 | 1.0475
$C^{*}_{T,a}$ | 1.1834 | 1.1624 | 1.1429 | 1.0985
$C^{*}_{P,a}$ | 1.1120 | 1.1010 | 1.0853 | 1.0660
$\eta^{*}$ | 1.0641 | 1.0558 | 1.0531 | 1.0306
$C^{\text{iso}}_{T,a}$ | 1.4937 | 1.4778 | 1.4622 | 1.4441
$C^{\text{iso}}_{P,a}$ | 4.5204 | 4.4757 | 4.4566 | 4.660
$u^{\text{iso}}$ | 0.0899 | 0.1043 | 0.1186 | 0.1473
Figure 8: (A) Trajectory of the follower with respect to the leader leading-
edge coordinates over 100 cycles for case IV. The initial positions that were
simulated are represented by the gray boxes. The color scale is mapped to the
dimensionless time. (B) The leader and follower positions as well as the wake
flows are highlighted at dimensionless times of $t/T=1$, $5$, $15$, and $60$.
The period of oscillation is $T$.

## Discussion and Conclusions

For the first time, we have discovered that a side-by-side arrangement of
pitching foils is not only two-dimensionally stable, but also two-
dimensionally super-stable where the relative distances between swimmers is
constant and the school as a whole does not have forces acting on it to drift
to one side or the other during locomotion. Indeed, the school is shown to be
super-stable for freely-swimming foils, which also enjoy modest speed and
efficiency gains over swimming in isolation. This provides new evidence that
the Lighthill conjecture (23) may play a role in school formation and spatial
patterns of swimmers even if only in a statistical sense akin to birds in a
V-formation (10). Indeed, it has been observed (11) that above a critical
swimming speed, tetra fish organize in a side-by-side lineup (or phalanx
formation as described in the study) and that they enjoy some energetic
benefit. Our findings support this previous work and provide a new viewpoint
that perhaps the self-organization of the tetra fish is passive in nature and
not an active control strategy by the fish.

While this is a provocative result, there are many differences between
schooling pitching foils and schooling tetra fish that make it difficult to
draw a direct conclusion about the fish. One important difference is that the
fish are composed of a body and fins, with it’s caudal fin undergoing a
combined heaving and pitching motion. However, recent work has shown that both
purely heaving and purely pitching foils experience one-dimensional stability
in an in-line arrangement (15), so perhaps the difference in kinematics is not
consequential for the existence of stable equilibria. Still, further work
should aim to examine the passive stability of truly fish-like swimmers to
directly answer this question.

Another complicating factor is that it has been argued that three-dimensional
swimmers in an infinite school experience breakdown of their shed wake
vortices (45), thereby disrupting the coherent vortex-body interactions that
drive the in-line and staggered arrangement interactions. The side-by-side
interactions, though, are dominated by oscillating dipole flow fields produced
from motions of the nearby bodies, which cannot break down like wake vortices.
This may mean that a stable side-by-side arrangement and its performance
benefits would hold even for three-dimensional swimmers and/or dense schools.
Further work in this direction is also warranted to provide some understanding
of how the two-body two-dimensional interactions of the current study
translate to many-bodied and three-dimensional interactions.

Previous work (25, 24, 27) has shown that in-line arrangements have multiple
one-dimensionally stable equilibria, while the current results show that those
equilibria, at least in the near-wake of the leader, are in fact unstable in
the cross-stream direction. However, this does not indicate that these one-
dimensionally stable points are irrelevant, but instead this work highlights
that the degree of stability falls along a spectrum. For instance, a fish
swimming in the wake of another fish may only need to actively control its
cross-stream position in order to maintain a schooling arrangement, which
requires less control effort than actively controlling two degrees of freedom,
but more effort than controlling none. While higher degrees of stability can
relieve needs for control strategies for swimmers and lead to completely
passive self-organization, lesser degrees of stability may more subtly sculpt
the schooling patterns observed in fish by influencing the trajectory
manifolds or the statistical positioning of swimmers. Beyond the translational
stability of arrangements, it is unclear whether the orientation of a swimmer
will also be stable to perturbations since it is unstable at least for some
synchronizations and arrangements (46).

Interestingly, when the thrust and efficiency performance are considered, the
ideal arrangement for maximizing performance is not the super-stable side-by-
side arrangement, though modest thrust and efficiency benefits can be reaped
in this arrangement. The optimal thrust and efficiency performance occurs for
a slightly staggered arrangement, which gives rise to interesting questions
about whether animals swim in energetically optimal arrangements through more
attentive control or in more stable arrangements with less performance
benefits.

This study provides a rich understanding of the interplay of stability and
performance in schooling pitching foils and these findings reveal hypotheses
for understanding biological schooling. These findings also provide insights
that aid in the design of multi-finned or schools of bio-inspired machines.

### Constrained Foil Experiments

Experiments were conducted for a minimal collective consisting of a pair of
hydrofoils constrained in space and immersed in a closed-loop water channel,
oscillating under prescribed sinusoidal pitching motions about an axis located
$8.4$ mm behind their leading edge. A constant flow speed of $U=0.093$ m/s was
imposed, which gives a chord-length based Reynolds number of $Re=9,$950\. The
flow over the hydrofoils was restricted to be nominally two-dimensional by the
placement of a surface and splitter plate near the hydrofoil tips. Two
identical hydrofoils with a rectangular planform, and NACA 0012 cross-section
were designated as the leader and follower. Each hydrofoil had a chord length
of $c=0.095$ m, and a span length of $s=0.19$ m, which gives an aspect ratio
of $AR=2$.
Particle Image Velocimetry measurements of the flow field were performed in
the horizontal ($X^{*}$-$Y^{*}$) plane at the mid-span of the foils. Phase-
averaged results were calculated from a total of 100 measurements for each
flow-field. A total of 16 distinct phases
($0,\frac{\pi}{8},\frac{\pi}{4},...\frac{15\pi}{8}$) were captured for all
schooling arrangements. The camera used was an Imager sCMOS (2560x2560
pixels), and seeding particles of 11 $\mu$m hollow metalic coated plastic
spheres were illuminated by a 200 mJ/pulse Nd:YAG laser. Multi-pass, cross-
correlation processing of the raw images was employed to obtain the resulting
vector fields, with a final interrogation window of 48$\times$48 pixels.

### Force Measurements

An ATI Nano43 six-axis force sensor was used to measure the thrust, lift and
pitching moment acting on each hydrofoil. An optical encoder recorded the
angular position information, which was then used to compute the angular
velocity, $\dot{\theta}$, for each hydrofoil. The total instantaneous power
input was then calculated as $P_{T}(t)=M_{\theta}\dot{\theta}$, where
$M_{\theta}$ denotes the pitching moment. Here, the inertial power was
determined from the same experiments conducted in air, and was subtracted from
the total power, $P_{T}(t)$, to calculate the instantaneous power input to the
fluid, $P(t)$. Force measurements were taken for 100 oscillation cycles from
the leader and follower, and each experiment was repeated 10 times. The time-
averaged values were calculated for each of these trials, and their mean from
10 trials was calculated to determine the time-averaged total thrust, lift,
and power. The profile drag was measured for the static foil in an imposed
flow over 20 seconds intervals. Net thrust was determined by subtracting the
profile drag acting on the hydrofoil(s) from the time-averaged thrust as
follows, $\overline{T}_{\text{net}}=\overline{T}-\overline{D}$. The
definitions of the net thrust, $C_{T}$, lift, $C_{L}$, and power, $C_{P}$,
coefficients and efficiency, $\eta$, are given as follows for the individual
hydrofoils:

$\displaystyle
C_{T}=\frac{\overline{T}_{\text{net}}}{\frac{1}{2}\rho{U}^{2}cs},\quad
C_{L}=\frac{\overline{L}}{\frac{1}{2}\rho{U}^{2}cs},$ (2) $\displaystyle
C_{P}=\frac{\overline{P}}{\frac{1}{2}\rho{U}^{3}cs},\quad\eta=\frac{C_{T}}{C_{P}},$

where $\rho$ is the fluid density and $s$ is the span-length of the
hydrofoils. Here, we also report collective performance parameters, that is,
the average performance from the leader and the follower. The collective force
and power coefficients as well as the collective efficiency are denoted with a
$C$ subscript and they are defined as,

$\displaystyle
C_{T,C}=\frac{\overline{T}_{L}+\overline{T}_{F}}{\rho{U_{\infty}}^{2}cs},\quad
C_{L,C}=\frac{\overline{L}_{L}+\overline{L}_{F}}{\rho{U_{\infty}}^{2}cs},$ (3)
$\displaystyle
C_{P,C}=\frac{\overline{P}_{L}+\overline{P}_{F}}{\rho{U_{\infty}}^{3}cs},\quad\eta_{C}=\frac{C_{T,C}}{C_{P,C}}.$

Note that, here, the performance coefficients were defined with combined
propulsor area, that is $2cs$, cancelling the one-half in the denominator.
Collective thrust and power coefficients, and efficiency were reported as
normalized values with the corresponding isolated hydrofoil performance metric
for comparison, and defined as follows,

$C_{T}^{*}=\frac{C_{T,C}}{2C^{\text{iso}}_{T}},\quad
C_{P}^{*}=\frac{C_{P,C}}{2C^{\text{iso}}_{P}},\quad\eta^{*}=\frac{\eta_{C}}{\eta^{\text{iso}}}.$
(4)

Here, the collective performance metrics are compared with the combined values
of two isolated hydrofoils. The isolated net thrust, drag, power and
efficiency are $C^{\text{iso}}_{T}=0.10\pm 0.015$, $C^{\text{iso}}_{D}=0.03\pm
0.002$, $C^{\text{iso}}_{P}=0.66\pm 0.0008$, and $\eta^{\text{iso}}=0.15\pm
0.022$, respectively.

### Unconstrained Foil Experiments

The unconstrained foil experiments were conducted to simulate a minimal school
of two swimmers where the follower is unconstrained (free-to-move) in the
$X^{*}$-$Y^{*}$ plane while interacting with the leader. Similar to the
constrained foil experiments, two identical hydrofoils with a rectangular
planform, NACA0012 cross-section, and aspect ratio $AR=3$ were designated as
the leader and follower. The leader hydrofoil was constrained in both the
streamwise ($X^{*}$) and cross-stream ($Y^{*}$) directions.

Similar to the constrained foil experiments, the foils were prescribed with
sinusoidal pitching motions by using a servo motor. The follower receives
start/stop commands from a computer via Infrared signal. Each servo motor is
controlled by an Arduino MEGA2560. A detailed description of the equipment and
data processing can be found in the Supplementary Material.

### Numerical Methods

To model the flow over a foil unconstrained both in the streamwise and the
cross-stream direction, we used a two-dimensional boundary element method
(BEM) based on potential flow theory in which the flow is assumed to be
irrotational, incompressible and inviscid. Previously, this method was used to
model flow over unsteady hydrofoils (47, 48, 39), and their interaction with a
solid boundary and the associated performance for constrained and
unconstrained foils (17, 28). Further details about the numerical method used
here can be found in previous work (39, 28).

Since the hydrofoils are self-propelled a drag force is applied to them, which
follows a high Reynolds number $U^{2}$ drag law (49). The drag force is
determined by the properties of a virtual body, not present in the
computational domain, which are determined by the Lighthill number, $Li$. This
dimensionless number is equal to $Li=C_{D}S_{wp}$, where $C_{D}$ is the drag
coefficient of virtual body and $S_{wp}$ is a ratio of the wetted area of the
virtual body and propulsor to the planform area of the propulsor. This
characterizes how the virtual body drag coefficient and the body-to-propulsor
sizing affects the balance of the thrust and drag forces. As previously shown
in (39), when $Li$ decreases at a fixed frequency and amplitude of motion,
free-swimming speed increases. This leads to a decrease in $St$ and
consequently, a decrease in the equilibrium distance as reported in (28). In
steady-state free-swimming the $Li$ is equal to the dynamic pressure based
thrust coefficient, that is $Li=C_{T}$. To investigate the stability of the
side-by-side equilibrium with recoil motions present, the Lighthill number was
then set to $Li=0.3$. This was the lowest value achievable for the numerical
stability of the current BEM formulation without wake impingement on the foil
bodies, which is a concern due to decreasing equilibrium distance between the
foils if the $Li$ takes lower values.

In self-propelled swimming the dynamic pressure based thrust coefficient
simply equals the Lighthill number. This is a fixed quantity in the case of
the simulations presented in the current study regardless of whether the
swimming speed increases or decreases. This means that the standard definition
of the thrust coefficient is meaningless in reflecting the thrust increase
that leads to higher swimming speeds when two foils are freely swimming side-
by-side. Thus, an added mass based thrust coefficient may be used as well as
an added mass based power coefficient for consistency (39). They are defined
as,

$\displaystyle C_{T,a}=\frac{\overline{T}}{\rho f^{2}A^{2}cs},\quad
C_{P,a}=\frac{\overline{P}}{\rho
f^{2}A^{2}Ucs},\quad\eta=\frac{C_{T,a}}{C_{P,a}}.$ (5)

The normalized added mass based thrust and power, and the normalized swimming
speed are,

$\displaystyle C^{*}_{T,a}=\frac{C_{T,a}}{C^{\text{iso}}_{T,a}},\quad
C^{*}_{P,a}=\frac{C_{P,a}}{C^{\text{iso}}_{P,a}},\quad
u^{*}=\frac{u}{u^{\text{iso}}},$ (6)

respectively.

This work was supported by the National Science Foundation under Program
Director Dr. Ronald Joslin in Fluid Dynamics within CBET on NSF award number
1653181 and NSF collaboration award number 1921809. Some of this work was also
funded by the Office of Naval Research under Program Director Dr. Robert
Brizzolara on MURI grant number N00014-08-1-0642.

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