The role of hydrodynamics in collective motions of fish schools and bioinspired underwater robots

Abstract

Collective behaviour defines the lives of many animal species on the Earth. Underwater swarms span several orders of magnitude in size, from coral larvae and krill to tunas and dolphins. Agent-based algorithms have modelled collective movements of animal groups by use of social forces , which approximate the behaviour of individual animals. But details of how swarming individuals interact with the fluid environment are often under-examined. How do fluid forces shape aquatic swarms? How do fish use their flow-sensing capabilities to coordinate with their schooling mates? We propose viewing underwater collective behaviour from the framework of fluid stigmergy , which considers both physical interactions and information transfer in fluid environments. Understanding the role of hydrodynamics in aquatic collectives requires multi-disciplinary efforts across fluid mechanics, biology and biomimetic robotics. To facilitate future collaborations, we synthesize key studies in these fields.

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royalsocietypublishing.org/journal/rsif
Perspective
Cite this article: Ko H, Lauder G, Nagpal R.
2023 The role of hydrodynamics in collective
motions of fish schools and bioinspired
underwater robots. J. R. Soc. Interface 20:
20230357.
https://doi.org/10.1098/rsif.2023.0357
Received: 23 June 2023
Accepted: 2 October 2023
Subject Category:
Life Sciences–Engineering interface
Subject Areas:
biomechanics, biomimetics, biophysics
Keywords:
fluid stigmergy, collective behaviour,
fluid mechanics, fish school
Author for correspondence:
Hungtang Ko
e-mail: hk1581@princeton.edu
The role of hydrodynamics in collective
motions of fish schools and bioinspired
underwater robots
Hungtang Ko
1
, George Lauder
2
and Radhika Nagpal
1,3
1
Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, USA
2
Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, USA
3
Computer Science, Princeton University, Princeton, NJ, USA
HK, 0000-0001-6250-6144; GL, 0000-0003-0731-286X; RN, 0000-0001-9756-0167
Collective behaviour defines the lives of many animal species on the Earth.
Underwater swarms span several orders of magnitude in size, from coral
larvae and krill to tunas and dolphins. Agent-based algorithms have
modelled collective movements of animal groups by use of social forces,
which approximate the behaviour of individual animals. But details of
how swarming individuals interact with the fluid environment are often
under-examined. How do fluid forces shape aquatic swarms? How do fish
use their flow-sensing capabilities to coordinate with their schooling
mates? We propose viewing underwater collective behaviour from the
framework of fluid stigmergy, which considers both physical interactions
and information transfer in fluid environments. Understanding the role
of hydrodynamics in aquatic collectives requires multi-disciplinary
efforts across fluid mechanics, biology and biomimetic robotics. To facilitate
future collaborations, we synthesize key studies in these fields.
1. Introduction
Animal collective behaviour has long been a fascination and inspiration for
humans [1]. Underwater swarms span several orders of magnitude in size,
from clouds of coral larvae on the millimetre scale to pods of dolphins on the
metre scale (figure 1). In groups, animals achieve functions that individuals
cannot achieve, forming a whole greater than the sum of its parts. Coordination
between animals in large groups requires unique strategies [2]. Biological organ-
isms have limited sensing capabilities and individuals do not know the precise
states (location, velocity, orientation, etc.) of other group members. How do
they respond to the sensory information available to them in a way that gives
rise to emergent behaviour on the collective level?
This question has been investigated by the use of agent-based models,
which simulate emergent group behaviour based on assumed rules for individ-
ual agents. Such models often impose a social force that either attracts or repels
an individual to its neighbour based on their distance as if they were connected
via a virtual spring (figure 2). Social forces like this are a crucial element of the
Boids model [3] and the Viscek model [4] as they facilitate generating coherent
movements in simulations. Despite its effectiveness, social force is only a
reduced-order approximation of animal behaviour, and limitations of animals’
sensing and locomotion are rarely considered (figure 2). While social force may
be fitted from experiments, there is no guarantee that its form and magnitude
apply to swarms of different sizes, densities and speeds: the social force is itself
an emergent property.
Going beyond such simplifications, the sensory systems of swarming
animals must be considered. Aquatic organisms perceive the world through
synthesizing a wide range of sensory information, including visual,
© 2023 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original
author and source are credited.

hydrodynamic, proprioceptive, electric and magnetic cues.
Classical models such as [5,6] were among the first to
consider a limited field of view. Others limited the
number of perceived neighbours [7], or designed agent
behaviours entirely based on vision [8]. The roles of sensing
modes other than vision have rarely been considered in
existing models of swarms. However, many aquatic species
have limited vision or live in dark environments. These
organisms cannot sense a neighbour a few body lengths
away as assumed in vision-based models. They must rely
on other sensory inputs like hydrodynamic cues and use
coordination strategies that are more short-range and
environment-dependent.
A crucial aspect often neglected in fish school models is
the effect of the fluid environment (figure 2). The concept
of environment-mediated coordination dates back more
than 60 years. Stigmergy was initially coined to describe the
collective construction of termite mounds [9–11]. Instead of
exchanging information explicitly with each other, termites
communicate and coordinate indirectly by leaving and
sensing traces in the environment. A similar principle is
also used by ants to follow trails [12], by paper wasps to
construct nests [13], and by honeybees to locate their
queens [14–16]. Recently, stigmergy has been extended
beyond social insects to bacterial colonies [17,18], spatio-
temporal patterns of animal territories [19], cognition
[20,21] and swarm robots [22–24].
In this perspective, we propose to view aquatic collective
systems through the lens of fluid stigmergy,whichurges
considerations of the dynamic fluid environment and the
information it carries. How are movements of underwater
collectives affected by fluid forces? How can individuals in
groups use hydrodynamic information for coordination? The
wake behind a fish carries information about the individual’s
state, much like the ants’pheromone trails. In addition,
spatial–temporal features of the vortex street are advected
(a)
(c)(d)
(b)
(e)
Figure 1. Underwater swarms in nature span several orders of magnitude, from (a) dolphins and (b) giant trevally on the metre scale, to (c) squid and (d)mackerel
on the centimetre scale, to (e) coral larvae on the millimetre scale. Images obtained through Education licences from Adobe Stock.
traditional models
sensing
states of
neighbours
muscle
actuation
locomotion
fluid
environment
fluid stigmergy
social
force
models
fluid–agent
interactions
lateral line
systems
other senses
e.g. vision
Figure 2. Different constructs of swarm models. Agent interactions in traditional models depend on relative distance and locations, sometimes considering vision
and a limited field of view. The perspective of fluid stigmergy includes the fluid environment and emphasizes fluid–agent interactions and coordination
strategies based on flow-sensing.
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2

and diffused according to the same principle that transports
airborne pheromones of social insects like honeybees [14–16].
This flow information is critical for underwater collective
movements, especially for animals with limited long-range
sensing capabilities. How stigmergy facilitates collective
behaviour likely depends on the scale and the Reynolds
number (Re), which defines the ratio between inertia and vis-
cous effects. For low Re organisms such as bacteria (Re <1),
disturbances diffuse almost instantaneously but may be
sensed from afar. For larger Re swimmers such as schooling
fish (Re =10
4
−10
6
), vortices retain their structures longer but
their influence is short-ranged. In this regime, while a swim-
ming fish is difficult to detect a few body lengths away, its
follower who swims through vortices may obtain rich infor-
mation about its states. Indeed, it has been demonstrated
that with only hydrodynamic information, a blindfolded fish
can school with its neighbours [25]. The relative role of
vision and flow sensing depends on the species and their
native habitats. But fluid stigmergy (figure 2) is likely used, at
least partially, by all schooling species.
Compounding the complexity of underwater coordi-
nation are the physical fluid–agent interactions. Passive
agents such as dead fish or granular materials can exhibit
non-intuitive behaviours under fluid forces. Just as hydro-
dynamic interactions can influence the collective behaviour
of microscopic organisms [26], repulsion and attraction
between a pair of fish can emerge in the absence of social
interactions and social forces [27,28]. In the wake behind an
obstacle, a dead fish can undulate its body and keep up
with the flow as if it were alive and swimming [29]. Recently,
it was shown that rheotaxic behaviour may not require any
sensory feedback either [30]. These hydrodynamic factors
led to the classical conjecture posed by Weihs & Lighthill,
which stated that a diamond formation of fish schools is opti-
mal because it would lead to constructive vortex interactions
and allow schooling individuals to save energy [31,32].
In addition, Lighthill surmised that such a formation is
stable—if a fish deviates from the diamond formation, fluid
forces will push it back to its original position. Whether a dia-
mond formation is truly hydrodynamically optimal and
whether fish schools in nature prefer such a formation
remains an open question. However, the key notion stands:
hydrodynamic interactions could lead to self-organizing
effects in spite of social interactions. Coupling with these
effects, movements of active agents can give rise to
unexpected collective behaviours [33].
Fluid–agent interactions affect not only natural swarms but
also underwater robot collectives. Recently, swarm robotics
have received increasing interest because they benefit from
low cost, versatility and robustness [34,35]. Most current
robot swarms are controlled using social force models.
Taking advantage of technologies such as WiFi, Bluetooth
and centralized tracking systems, terrestrial robots can per-
ceive neighbours’locations better than biological organisms,
making coordination simpler. However, underwater robots
do not enjoy the same benefits. Electromagnetic waves attenu-
ate rapidly and sonar suffers from low bandwidth and high
latency [36]. As such, low-cost and low-power underwater
swarms must rely on visual and hydrodynamic cues much
like schooling fish. Recently, a vision-based underwater robot
swarm has been demonstrated to achieve collective behaviours
such as milling [37]. Future designs that include a flow sensing
system may enable robot swarms to save energy by taking
advantage of the fluid–agent interactions and using fluid
stigmergy for coordination.
In this review, we discuss findings and methods in fluid
mechanics, biology and robotics with the perspective of
fluid stigmergy. We limit our scope to underwater systems
since most aerial animals such as starlings possess acute
vision and unmanned aerial vehicles often rely on wireless
communication. In the next section, we start by examining
the fluid field around fish schools, discussing what hydro-
dynamic information a swimming fish leaves behind, and
how swarming individuals may interact with each other
through fluid forces. In §3, we discuss what is known
about how fish sense their fluid environment. In §4, we
review robotic platforms with flow-sensing capabilities and
summarize what behaviours flow-sensing enables. Finally,
in §5, we highlight key challenges and opportunities for
understanding underwater systems in the future.
2. Understanding fluid flow
To start, we must understand the fluid mechanics of fish
schools. What is the flow signature a fish leaves behind?
How is that information preserved and propagated to
another fish nearby? What schooling formations are hydro-
dynamically efficient and stable? The complex interaction
between fish schools and water flow has been an ongoing
research topic within the fluid mechanics community, and
progress has been made using both experiments and
simulations [38–41,42, ch. 3].
When a fish swims through the water, it leaves two rows
of vortices behind (figure 3a). Each vortex forms as the tail
swings from one side to the other and sheds when it reverses
course. Vortices alternate in their direction (clockwise versus
counterclockwise). As they travel downstream, vortices dissi-
pate energy and break down into smaller eddies. The rate of
energy dissipation and the angle of the vortex street depends
on various factors such as the swimming velocity and the
flow regime (turbulent versus laminar). The width and wave-
length of the wake decrease as the tail-beat frequency
increases and smaller vortices develop when the frequency
is low [46]. The flow signature behind a swimming fish has
a strong frequency component. Recently, it has been shown
that fluctuations of the fluid field preserve information
about a fish’s relative position, phase differences and tail-
beat frequency [44].
The simplest way to generate a wake is by placing an
obstacle in a current. The flow over a fixed blunt object
produces alternating vortices, called the Kármán vortex street
[47](figure 3a). This flow pattern can be found behind pillars
under bridges or behind rocks in rivers. The rotation directions
of the vortices in Kármán vortex street are the opposite of
those in a fish’s wake. Therefore, the wake of a fish is also
referred to as the reverse Kármán vortex street. Fundamentally,
the arrangement of the wake vortices relates to whether the
object is producing drag (pillars) or thrust (fish) [48]. A key
parameter that determines the force balance is Strouhal
number, St =fL/U, or frequency fnon-dimensionalized by
the length scale Land flow velocity U. The emergent vortex-
shedding frequency of a Kármán vortex street behind a cylin-
der results in St ≈0.2−0.3. Remarkably, for fish swimming at
higher speeds, St is also in this same range [38,39,41,49].
Now defined with tail-beat amplitude as the length scale
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and swimming speed as the velocity scale, the non-dimen-
sional tail-beat frequency St ≈0.3, with most aquatic species
ranging narrowly within 0.2–0.4 [38,39,41,49]. This relationship
can be used to estimate an undulatory swimmer’s speed based
on its beating amplitude and frequency.
Fish wakes can be further abstracted using the potential
flow theory in fluid mechanics [50]. Flow around a cylinder
can be expressed analytically as a dipole. Since the theory
neglects viscous effects, the flow around a dipole does not
shed periodic vortices and only approximates the time-
averaged fluid field (figure 3a). To better mimic the wake
behind a fish, in both experiments and simulations, research-
ers use oscillating spheres and dipoles. However, even in
such a simplified setting, complexity quickly arises when
there is more than one body in the fluid domain. Placing a
sensor or a fish body (dead or alive) will instantaneously
change the flow field, including the structure of the vortex
street. Indeed, the hydrodynamic interactions between a
fish body and its own fins can already lead to non-intuitive
results [51,52]. To study what a fish senses from its neigh-
bour’s wake, it is crucial to include multiple fish in the
fluid field.
The hydrodynamics of fish schools remain an active
area of research. The elegant conjecture that Weihs and
Lighthill proposed about the diamond formation (figure 3b)
has been challenged with animal experiments [53–55], and
numerical simulations [27,56,57]. Weihs’argument for a dia-
mond pattern is based upon constructive vortex interaction in
directions both parallel and perpendicular to the swimming
directions, assuming the formation remains unchanged.
However, a recent study suggested that the constructive
fluid–structure interaction is more reliable in the fish’s lateral
direction [56]. Therefore, the side-by-side ‘phalanx’formation
is more efficient than the diamond formation (figure 3b). This
claim is supported by experiments of schools of red nose tetra
Hemigrammus bleheri, who form increasingly one-dimensional
patterns as swimming speed increases. This phalanx for-
mation allowed the tetras to reach faster swimming speeds
with lower tail-beat frequencies [55,58]. However, exper-
iments with five species of rainbowfish (Melanotaenia)
arrived at a conflicting conclusion, showing more alignment
along the swimming direction (figure 3b) as their speed
increases [59]. To add to the puzzle, using two-dimensional
simulations with periodic boundaries, research suggests
that hydrodynamic benefits are easier to obtain than pre-
viously thought [57]. Fish schools are more efficient than
individuals regardless of the formations (diamond, rectangu-
lar, side-by-side and inline) [57]. It is unclear if and how the
preferred schooling formations depend on species, especially
regarding the relative importance of the senses (hydro-
dynamic versus vision) for coordination. Perhaps more
critically, whether fish can maintain fixed formations for pro-
longed periods at all is questionable. Both early literature and
empirical observations suggest that fish school formations are
transient and dynamic [60].
How do fluid forces push fish in and out of transient for-
mations? Investigating the hydrodynamic stability of various
formations has gained considerable interest. For a pair of
hydrofoils in tandem with a constant frequency, amplitude
and spacing, two stable swimming speeds exist [61]. The
two emergent modes, fast and slow modes, correspond to
when the flappers are out-of-phase and in-phase, respectively.
If the spacing is allowed to vary spontaneously, configurations
with synchronized phases (slow modes) are always preferred
and are hydrodynamically stable—fluid force will restore
their states if perturbed [62]. As more degrees of freedom are
added, a stability diagram may be obtained [63]. When the
frequencies of the two hydrofoils are matched, stable configur-
ations exist for a range of heaving amplitudes. Intriguingly,
an oscillatory mode also exists in the regime where the follower
streamlines around a dipole
Kármán vortex street
wake behind a fish
diamond
inline
phalanx
(a)
(d)(e)
(b)
(c)
Figure 3. (a) Fluid fields behind a swimming fish, behind an obstacle, and around a dipole. Red vortices rotate clockwise while blue vortices rotate counterclock-
wise. (b) Hydrodynamic arguments support either a diamond formation, a phalanx formation or an inline formation. (c) The three-dimensional fluid field around a
swimming fish can be characterized by tomographic PIV (permission from [43]), and hydrodynamic interactions between a pair of fish can be studied using (d) fluid
simulations (permission from [44]) and (e) hydrofoil experiments (permission from [45]).
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4

heaves wider but less frequently. In this mode, fluid forces
orchestrate the periodic movement of the follower, at times
closing the gap and at others failing to keep up. These beha-
viours have been observed in numerical simulations where
the swimmers deform their bodies like fish and are allowed
to rotate [64,65]. Oscillatory modes have also been found for
two simulated fish starting from side-by-side formations [66].
For up to four swimmers, researchers have discovered more
than a dozen equilibrium formations, including the diamond
formation that Weihs suggested [67]. However, simulations
in [67] are two-dimensional, and swimmers can only move
freely along the streamwise direction. Recent studies using
three-dimensional simulations [27] and hydrofoils free to
move in both directions [28]showedthatveryfewformations
are truly stable.
These studies of hydrodynamic performance and stability
of fish schools have generated considerable insight into
schooling strategies. The fact that stable equilibrium states
are rare suggests a stability–manoeuvrability trade-off
similar to bird flight [68,69]: if fish are attracted strongly to
equilibrium positions by fluid forces, they will lose the
ability to change directions rapidly. However, while a bird
can engineer the stability landscape by changing its body
conformation and movement frequency, the stability land-
scape of a fish is determined by the movements of its
schooling mates! The scarcity of stable formations also
suggests that realistic formations of underwater swarms
must be ever-changing (unless deliberate efforts and costs
are paid to maintain them). This, however, does not rule
out the possibility of identifying any patterns in transient
schools since dynamic oscillatory modes are also likely.
To study the hydrodynamics of fish schools, a wide range
of experimental and numerical tools may be adopted. Particle
image velocimetry (PIV) is widely used in experiments to
characterize fluid flows around live fish [70–73]. The technique
has been extended to study the three-dimensional flow field
around individual swimming fish (figure 3c)[43,74–76].
PIV has also been applied to multiple interacting hydro-
foils (figure 3d)[28,45,61,62]. Numerically, researchers
have used methods such as direct numerical simulations
with immersed boundaries (figure 3e)[27,44,56,66,67,77–79],
multi-particle collision dynamics model [57], and potential
flow based solvers [50,80]. However, most existing simulations
prescribe the movements of the fish bodies, and the force
balance around a fish is not always strictly enforced. As
these computational techniques advance, realistic simulations
with multiple interacting individuals may soon be achieved.
Further, interfacing fluid simulations with agents that can
actively sense and react to their fluid surroundings is a fruitful
direction for future work.
3. Flow-sensing in fish schools
The sense of touch (mechano-sensation) is universal for ani-
mals both on land and underwater, but the form of
mechanical forces differs significantly. Aquatic animals
adopt unique strategies to sense fluid flows around them
and can sense ‘touch at a distance’[42]. Their mechano-
sensory organ, the lateral line system, senses perturbations
across a wide range of frequencies, from steady currents
to acoustic waves. Lateral line systems have attracted signifi-
cant research since the seventeenth century and have
been reviewed excellently by Coombs and co-workers
[42,81]. Here, we focus on the mechanical properties of lateral
line systems as they relate to the function of sensing fluid
movement and their role in implicit coordination.
Lateral line systems can be divided into two subgroups:
superficial neuromasts (SN) and canal neuromasts (CN)
(figure 4a). They consist of dome-shaped epithelial structures
(cupula) covering hair cells that deflect under fluid flows.
Both types of neuromasts sense fluid shear to infer flow
intensity. SNs protrude from the surface and directly sense
the flow velocity in their vicinity. CNs sense fluid flow
within the lateral line canal. Since the flow in the canal is
dominated by viscous effects, its speed is proportional to
the pressure difference between pores that are connected
to the exterior environment. Therefore, CNs effectively
sense the pressure drop in the external flow. SNs often
locate in proximity to CNs [83], and canals can be found
both on the head (cranial canals) and along the body
of a fish (trunk canal ). Canals are denser on the head, sur-
rounding the eye and extending from the cheek to the jaw
(figure 4b), providing hydrodynamic velocity information
from all directions. The difference among those signals may
be used to infer the direction of flow or to locate prey
[82,83]. With only two types of fluid sensors available to
them (SNs and CNs), how do fish arrange them to maximize
their sensing capabilities? The placement and relative abun-
dance of SNs and CNs are extremely diverse (figure 4b)[42,
ch. 2; 81, ch. 10; 82–84], but it has been noted that fish
living in fast and turbulent waters tend to have only a few
or no SNs [82]. This trend may be explained by considering
the boundary layer that forms around the body of a fish
(figure 4a)[73,85–87]. The boundary layer is thinner for fast
and turbulent flows. In these environments, the boundary
layer can shrink to thicknesses comparable to the size of neu-
romasts, and the protrusion of SNs would significantly alter
the flow. Sheltered by the canal, the performance of CNs is
less affected by boundary layers in fast currents.
Hydrodynamic information produced by a moving fish is
intrinsically periodic. While a single tail beat only creates
small disturbances that dissipate rapidly with distance and
time, periodic tail beats produce pressure waves that can
travel much farther than the scale of the animal. Indeed,
swimming fish create acoustic waves! It is unclear if such a
wave is strong enough to be distinguished from background
ocean noise, but research has long shown that fish can use lat-
eral line systems to sense low-frequency vibrations [42]. In
fact, during most of the nineteenth and twentieth centuries,
lateral lines were considered auditory organs, and researchers
have investigated how neuromasts respond to oscillatory
flow [42]. Like trees and buildings, neuromasts have natural
frequencies, and their oscillatory movements depend on iner-
tia and rigidity. As a result, signals within a certain frequency
range are amplified, while others are attenuated (figure 4c).
Cave-dwelling populations of Mexican cavefish (Astyanax
mexicanus) have taller SNs and are more responsive to
lower frequency stimuli compared with stream-dwelling
populations that possess functional visual systems [88].
Fluid properties also affect neuromasts’frequency response.
As the perturbation frequency increases, inertial effects
begin to dominate over viscous effects, and boundary
layers form. This can be explained by Stokes’s second pro-
blem, which suggests that the fluid shear next to SNs
increases with the square root of frequency, ffiffi
f
p. For CNs,
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the channel flow profile also depends on the frequency,
adding a third layer of filtering. Astonishingly, the combi-
nations of frequency filtering result in a constant sensitivity
to flow velocity in SNs of zebrafish (Danio rerio)[89] and a
constant sensitivity to flow acceleration in CNs of ruffe (Gym-
nocephalus cerua)[90] for a wide range of frequencies. This
indicates that SNs and CNs are indeed suitable for sensing
velocity and pressure, respectively.
Flow sensing of the lateral line systems enables beha-
viours such as prey detection, obstacle avoidance and
rheotaxis [81, ch. 3]. Early studies focused primarily on
how lateral line systems sense perturbations from small
prey. Nocturnal predators such as the Lake Michigan mottled
sculpin (Cottus bairdi) rely heavily if not exclusively on the lat-
eral line [91,92]. Species that typically use vision to locate
prey, such as bluegill sunfish (Lepomis macrochirus)[93], large-
mouth bass [94] and blacktip sharks [95] can still detect prey
without vision as long as the prey is within half a body length
away. Surface-feeding fish use lateral line systems to detect
the pressure variation caused by prey at the water surface.
Blind cavefish Astyanax mexicanus rely on hydrodynamic sen-
sing to follow walls [96,97], avoid obstacles [72] and detect
prey [88]. Sensing flow allows fish to hold their position
either in free streams as they orient towards the incoming
flow [98,99] or in regions behind an obstacle [100–102].
More recent studies have focused on the role of hydro-
dynamic sensing in dynamic manoeuvres. Kármán gaiting
describes a synchronized movement between a fish and the
vortex street behind an obstacle drag wake [103,104].
In such a gait, fish slalom in between the shed vortices to
take advantage of the propulsion created by the fluid
pressure gradient. This behaviour is only possible when
fish can access the local hydrodynamic information [100].
Sensing vortex streets also allows European catfish (Silurus
glanis) to follow the hydrodynamic trail of their prey
[105,106]. Recently, vortex-matching has been identified as a
coordination strategy for fish in pairs [46,107,108]. Surpris-
ingly, it was concluded that neither vision nor lateral line
systems are required for such behaviour [108], suggesting
that it can arise from hydrodynamic interactions alone.
Only a few studies exist on the role of hydrodynamic sen-
sing during schooling. Classical experiments were conducted
in the late 1970s. Pitcher & Patridge investigated how a fish in
a school behaves differently when it is deprived of vision or
fluid sensing capabilities [25,109]. They showed that a blind-
folded fish can still school, but with a different structure
[109]. When vision is deprived, the fish prefer to stay closer
to their neighbours. They concluded that social attraction,
i.e. the active strategy to cohere as a group, is mediated by
vision, whereas collision avoidance is mediated by lateral
line information. Recent studies on firehead tetras (Hemigram-
mus bleheri), yellow-eyed mullets (Aldrichetta forsteri) and
giant danios (Devario aequipinnatus) also showed that lateral
line ablation disrupts schooling [110–112]. Without lateral
line information, fish align less and collide more with other
fish [110,111]. More recently, researchers demonstrated that
while vision is sufficient for schooling in rummy-nose tetras
(Hemigrammus rhodostomus) as they burst and coast in still
water [113,114], the lateral line systems in giant danio (Devario
aequipinnatus) is crucial for tail synchronization [115]. Exper-
imental evidence also demonstrated that giant danio
(Devario aequipinnatus) can form coherent schools in a dark
room [116] and coordinate with a flapping hydrofoil or a
robot that produces fish-like wakes [117,118]. It appears
unquestionable that a certain level of schooling can be
achieved in the absence of visual cues.
In addition to vision, inner ear hair cells sense the
acceleration of the animal and the proprioceptors in muscles
and connective tissues detect body deformation and fin
deflection. These sensory modes provide supplemental
information on the fluid environment that fish may use
for coordination. Future animal experiments are required
(a)
(b)
SN
CN
(c)fluid filtering structural resonance
superfacial
canal
low freq. high freq.
low freq. natural freq.
frequency
frequency
deflection/local flow velocity
local flow velocity/freestream velocity
frequency
deflection/freestream velocity
high freq.
CN
SN ~f2
const. response to acceleration
const. response to velocity
Figure 4. (a) Superficial neuromasts (SN, red) and canal neuromasts (CN, blue) and their relative relationship with the boundary layer flow. Grey arrows indicate
fluid velocity. (b) Interspecies diversity in trunk canal placements (adapted from [82]). (c) Frequency response of SN and CNs (adapted from [42]).
royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 20: 20230357
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to elucidate the role of lateral line systems in schooling
behaviour.
4. Flow-sensing in underwater robots
Inspired by lateral line systems, engineers have developed var-
ious electronic platforms that sense flows. These mechanical
systems provide powerful tools to study fluid-mediated coordi-
nation in both natural and artificial systems. Fluid flow around
fish schools depends only on the boundary conditions. There-
fore, by mimicking the shape and movement of fish, a robot
can reproduce the same wake, and sensors placed at locations
of real fish’s lateral line systems will receive the same hydrodyn-
amic information a fish would access. Hypotheses about how
fish coordinate with each other can then betested. Furthermore,
incorporating flow sensing into underwater robot swarms
allows artificial systems to adopt fish behaviour such as
rheotaxis, obstacle avoidance and coordination.
On the smallest scale, lateral line systems inspire the
designs of miniature flow sensors called ALL (artificial lateral
line) sensors [119–122]. Taking advantage of recent advances
in MEMS and NEMS (micro- and nano-electro-mechanical
systems) manufacturing, new designs of flow sensors mimic
structures of SNs and CNs and often include a single or an
array of micro-pillars. Micro-pillars bend under fluid flow
much like SNs, and piezoresistive, piezoelectric, capacitive
or optical components can be used to convert mechanical
deformation into electrical signals [120]. Since the same
mechanism was used, these SN-mimicking sensors also
respond differently to stimuli at different frequencies. In the
past few years, there has been an increasing interest in devel-
oping CN-inspired sensors that measure the pressure
difference between canal outlets [40]. Like in fish, artificial
CNs are more suitable compared to artificial SNs when the
boundary layer is thin. Protected by microfluidic channels,
CN-inspired sensors are also less prone to damage. While
ALL sensors closely mimic the mechanisms of lateral line
systems and are customizable, they are difficult to manufac-
ture and often require additional tuning and calibration.
Therefore, most existing designs of autonomous robots use
commercial flow sensors instead of ALL.
Beyond bio-mimicry, robotics has emerged as a powerful
tool for generating and testing biologically relevant hypotheses.
BlueSwarm is the first autonomous underwater robot swarm
capable of decentralized coordination (figure 5a)[37]. It demon-
strates that robot fish can display milling behaviour by using
vision alone. Nevertheless, the robots are limited to still
waterand slow speeds, where flow sensing may play a less sig-
nificant role. One of the earliest robot fish with hydrodynamic
sensing capabilities is the prototype under project FILOSE
(Robotic FIsh LOcomotion and SEnsing) [123,127–130]. In
2011, Kruusmaa et al. reported how a robot fish could use one
single pressure sensor to control its tail-beat frequency to keep
up with the flow speed in the water channel [123](figure 5b).
By incorporating a controller inspired by the Braitenberg
vehicle, this robot can perform rheotaxis by sensingthe pressure
difference between the two sides of its head [128]. With all
five pressure sensors activated, the robot fish can differentiate
between the flow behind an obstacle, i.e. Kármán vortex
street, and an unobstructed laminar flow. In both cases, it is
capable of exhibiting station-holding behaviour using
hydrodynamic sensory information [130].
Sensing robots without fish-like beating tails benefited from a
higher signal-to-noise ratio and enabled theoretical analyses.
Researchers have increased the number of pressure sensors
from 9 [124,131], 11 [132,133]to20[134](figure 5c), making
them effective flow sensing probes for measurements in the
field [132,133]. The robotic platforms developed in [135–137]
also lack actuated tails. Using a pairof Joukowski foils constrained
to planar translation and pitching, researchers implement
model-based control for the follower to perform Kármán gaiting
[137]. Untethered robots can also benefit from sensing the fluid
environment. Flow sensing enables free-swimming robots
to estimate the robot’s own swimming speed and direction
[138,139] and to follow the wall like a cavefish [140,141]. An eel-
like underwater robot has been developed to coordinate its
body undulations based on hydrodynamic feedback [142].
Flow sensing has also been used in robots to gather infor-
mation about their neighbours. For a pair of robots towed in a
leader follower
flow sensors
10.4 cm
(a)(c)
(d)
(b)
(e)
12
3
4
5
Figure 5. Underwater robots with onboard sensing capabilities. (a) Vision-based coordination in BlueSwarms, reproduced with permission from [37]. (b–e) Indi-
vidual robots with flow sensors (permission from [123] for b, and [124] for c). (d,e) Robots that use flow-sensing to detect their neighbours (permission from [125]
for d, and [126] for e).
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7

flume, the follower robot can use the sensed hydrodynamic
pressure variations to estimate the states of the leader
robot, including its relative location, orientation, oscillating
frequency, amplitude, etc. (figure 5d)[125,143]. A free-
swimming robot can also locate its leader (figure 5e)[126].
The estimation of the neighbour’s location is more accurate
when the neighbour is in front of the follower compared to
when the neighbour is beside or behind the robot [126].
These studies have suggested that hydrodynamic information
can indeed be used for coordination. Furthermore, they show
the potential of using robots as a powerful tool for extracting
hydrodynamic information and using it to achieve fish-like
behaviour. However, free-swimming robots are rare and
most studies are limited to only two sensing robots.
Adding artificial lateral lines to platforms like BlueSwarms
[37] remains an exciting future direction.
5. Challenges and opportunities
Understanding underwater collectives requires knowledge
from various disciplines. Research using live animals and
robots presents two distinct philosophies for tackling
complexity. In animal research, it is often difficult to control
and isolate variables of interest, leading to an often messy
but realistic picture. On the other hand, a biomimetic
approach builds the complex system from the ground up
and is excellent for studying each contributing factor in
isolation. However, constructing these underwater systems
is challenging and the fidelity of bio-mimicry is always com-
promised to some extent. Future studies that join different
approaches would reveal novel insights into underwater
collectives (figure 6).
5.1. Building realistic models of swarms
Agent-based models are critical for exploring and under-
standing collective phenomena, but the dynamics of the
fluid environment has often been ignored. While this may
be valid for vision-dominated animals such as starlings
[144–147], underwater organisms have to interact with
much larger fluid forces and the visibility of their environ-
ment is often lower. To investigate underwater systems, it is
essential to couple agent-based models with computational
fluid dynamic simulations (figure 6a). Such a coupled
model will capture the realistic interaction between agents
and their fluid environments: agents’movements disturb
the fluid field, and the fluid in turn propels the agents. Incor-
porating fluid field simulations also creates the opportunity
to model agents’behavioural rules based on their fluid sur-
roundings. Such sensory-based multi-agent models will
enable studies of how agents react to hydrodynamic cues,
and how variations of this feedback mechanism affect the
cohesiveness of the school (figure 6b).
The most glaring challenge with coupling collective
behaviour to a flow solver is the computational cost. Current
advances in computational fluid dynamics allow resolutions
of fine details in flow structures. However, high-fidelity
algorithms require a considerable amount of computational
resources and time. One direct numerical simulation of a
fish school swimming for a few seconds can take weeks to
complete on a computer cluster. To reduce the computational
cost, most existing fluid simulations of fish schools prescribe
the fish’s movements and ignore the force balance around
each individual and Newton’s second law. If the prescribed
conditions such as the formation, tail-beat amplitude and
frequency are inherently unstable, the formation cannot
be sustained and any solutions reached would not be
representative of biological collectives.
To ensure a force balance around individual agents while
keeping computational cost low, the fidelity of the simulations
must be compromised. This can be done by using potential
flow theories [50,148]. However, since agents under these
assumptions do not leave periodic wakes, the treatment is
only applicable for sparse underwater collectives. An alterna-
tive method is by directly modelling the vortex street behind
a two-dimensional swimmer [149]. In addition, adaptive
mesh refinement and simplified agents have been used to con-
duct a parameter study of insect swarms under airflow [33].
Another promising approach that does not require significant
sacrifice in flow field accuracy is to prescribe the trajectories
theory
pressure
explore agent rules based on
hydrodynamic and visual information
school cohesiveness
agents’ reactivity to information
(a)
(b)
biology
characterize large three-dimensional
schools of freshwater and marine species
tracks of individuals
time
probability in diamond formation
time
(c)
(d)
robotics
design flexible autonomous fish-like
robots that can sense flow in fish schools
time
sensed pressure
in school
free swimming
(e)
(f)
Figure 6. Future directions and goals for understanding underwater collectives using (a,b) theoretical, (c,d) biological and (e,f) robotic approaches.
royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 20: 20230357
8

and kinematics tracked from animal experiments. This
approach will ensure that the simulation satisfies physical con-
straints automatically. The force balance around individuals in
the simulation will be left as a check for numerical accuracy.
Striking a balance between the computational cost associated
with fluid simulation and reality (deformable body, uncon-
strained movement, large number of individuals, behavioural
complexity) is critical for future research.
5.2. Characterizing collective movements through
experiments
One fundamental challenge of studying fish schools is
the difficulty of conducting animal experiments. Keeping,
experimenting with, and tracking fish schools in three dimen-
sions even in a laboratory environment requires expertise in
both biology and computer vision (figure 6c). Most existing
experiments used small freshwater fish (a few centimetres
in length) in shallow aquaria, forcing formations to be
planer which made tracking manageable [55,150]. While
much progress has been made in animal pose tracking algor-
ithms using approaches such as contrast-based segmentation
[151–153] and machine learning [154–156], their performance
in tracking dense fish schools remains less than ideal. Using
one of the most popular tools in multi-animal tracking
in DeepLabCut, tracking near planar shoals of silversides
Menidia beryllina yields at best 70% mean average precision
(mAP) [155]. For schools in three dimensions, fish frequently
overlap with each other in any given video view. Isolating
individuals from overlapping clusters requires ingenious
solutions. Future tracking algorithms may benefit from
using multiple cameras and stereo vision to simultaneously
acquire a depth map from the camera [52,115], or from incor-
porating knowledge of fish swimming kinematics which have
been reported to be generally similar across a diversity of
fish families [157]. The recent development of large models
for general computer vision such as [158] also provides
tremendous opportunities for three-dimensional tracking
fish schools.
To gain a deeper understanding of underwater collective
behaviour, studies of a wider range of species are also critical
[159]. Our knowledge of marine species that migrate in
enormous groups, such as tuna and sardines, remains limited.
Field experiments and novel approaches are essential. Using
a towed net, a recent comparative study examined the swarm-
ing patterns of California market squid (Doryteuthis opalescens)
and Pacific sardine (Sardinops sagax), which have distinct pro-
pulsive mechanisms [160]. Using rectified videos, another
team of researchers measured the duration and size of fish
shoal disturbances [161]. Adaptive resolution imaging sonar
(ARIS) [162,163] and drone footage [164–167] have also
emerged as powerful tools in identifying schooling structures
in the field.
Studies of underwater swarms can also benefit from rig-
orous characterizations of their spatial–temporal features.
Many influential studies that shaped our understanding of
schooling formations use experiments that are based on
trials under a minute long [55,59]. At this timescale, the
habituation of the animals and the initialization of flow
field may both affect the schools’structures. Recordings
that are around an hour long allowed researchers to apply
statistical methods and analyse the state of the school
[150,168]. Even longer recordings that are multiple hours or
even days long are necessary to fully characterize fish
school formations and answer fundamental research ques-
tions: How frequent are fish schools in certain formations
(figure 6d)? How stable are these formations? It will be inter-
esting to see if other patterns of coherent movement will
emerge from systematic studies like these.
Tracking fish schools provides a crucial step toward under-
standing their collective behaviour. Since the flow field
depends only on the movement of the school, it can be recon-
structed using fluid simulations or robotic experiments once
the tracks are obtained. Through such hybrid approaches,
one can obtain detailed flow information such as the fluid
stress and pressure field next to each individual’s lateral line
systems without directly experimenting with the animals. In
addition, other sensory information such as vision and
proprioception may also be inferred. Without further measure-
ments, it will be tangible to derive a mapping from what a fish
may sense to its behaviour. Note that both the input and
output spaces of this mapping are large and the relationship
may be nonlinear. Searching for low-dimensional structures
from high-dimensional data is a recurring theme in biophysics
[169]. Dimension-reduction techniques such as principal
component analysis and proper orthogonal decomposition
may be used [170,171]. Statistical methods such as correlation
functions and transfer entropy may also be applied [117,172].
Furthermore, future research may adopt cutting-edge
model-discovery approaches such as sparse identification of
nonlinear dynamical systems (SINDy) [173] and knowledge-
based neural ODE (KNODE) [174,175]toinferthebehavioural
strategy of schooling fish. Application of these methods
on thousands of tracks in a large school is an exciting
future direction.
5.3. Designing flow-sensing robot collectives
Bioinspired robots enable rigorous studies of hydrodynamic
effects on underwater swimmers, isolating factors related
to animal behaviour. While tethered robots and hydrofoils
provide better control for experimental work, they prevent
natural movements that arise from hydrodynamic inter-
actions. Air bearing systems allow a tethered object to move
freely in two dimensions, but movements in the other four
degrees of freedom are constrained. Critically, the pitch
angles of hydrofoils are often controlled. In fact, rigid hydro-
foils can only generate thrust by pitching, which is driven by
the motor it is attached to. This is distinct from fish, who gen-
erate thrust by contracting their muscles and deforming their
body. In the absence of external control, free-swimming
robots have to generate thrust through deformation and
thus move more like real fish [37,176–178]. It has also been
discovered that tuning flexibility is essential to match the kin-
ematics of fish [177,179–183]. Future studies that deploy
groups of autonomous free-swimming robots may lead to
discoveries of metastable schooling configurations that
cannot be observed with tethered hydrofoils.
Bioinspired robots also provide unique opportunities to
peek into the sensory world of biological organisms. If the
morphology and kinematics of fish are mimicked, a fish
robot in a swarm will sense the same fluid pressure and
shear that a fish in such a school must sense. While electronic
sensors may differ from neuromasts in their sensitivity and
range, the hydrodynamic information accessible to them is
identical. Adding arrays of flow sensors to systems like
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9

BlueSwarm [37] would reveal detailed information about the
flow field around the fish robot. Hybrid experiments that
swim fish robots among living fish school will provide
insights into the roles of sensing in coordination and school
formation control (figure 6e,f). With the added flow-sensing
capabilities and knowledge of fish’s hydrodynamic feedback
algorithm, future robots may be able to perform vortex
matching and increase energy efficiency. Development of
miniature artificial lateral line sensors that are scalable and
easily deployable will also significantly advance our under-
standing of the hydrodynamic information embedded in
the environment.
6. Concluding remarks
Fish schools are mesmerizing. They have inspired people
from all walks of life and are an excellent example of collec-
tive behaviour. They are also easy to observe; most pet stores
sell dozens of tropical species that readily school in aquaria.
Despite the ubiquity, it remains mysterious how they coordi-
nate with each other so well and how fluid environment
dictates their collective behaviour.
To some, the mystery may seem to have been solved
when the Vicsek and Boids models showed that by keeping
a certain distance from their neighbour, a flock of agents
could avoid obstacles and move cohesively. Nonetheless,
these models overlooked limitations of sensing in biological
organisms. The range, resolution and speed of information
acquisition are all limited; an animal cannot perfectly esti-
mate its neighbours’locations and their distances from it.
This gap in our understanding of swarms becomes glaring
as roboticists struggle to translate the social force models to
robots—robots also have limited sensory capabilities.
In bridging the gap, interdisciplinary collaboration is
essential. Fluid mechanics researchers have long pondered
the fluid–structure interactions of hydrofoils; biologists have
meticulously described the morphology and sensing mechan-
ism of lateral line systems; roboticists can now assemble fleets
of underwater vehicles. To gain deeper insights into under-
water coordination, we must gather a diverse team of
experts from different backgrounds.
Finally, we look forward to applications of fluid stigmergy
in other collective systems where signals are transported by
dynamic fluid fields. In fact, many organisms from micro-
scopic to macroscopic coordinate with each other using
chemical signals that are suspended in fluids like water or
air. The role of fluid flow in these collectives remains to be
investigated.
6.1. Citation diversity statement
This statement is inspired by a recent initiative [184]. The accu-
mulated knowledge in the topic reviewed here joins not only a
diversity of disciplines but also contributions from researchers
of different genders and ethnic origins. Recent work revealed
that women and other minority scholars are under-cited even
as factors representing seniority and productiveness are
accounted for (see [185] for a detailed analysis). Here, we pro-
vide a breakdown of our references using the tool https://
github.com/dalejn/cleanBib. Our references contain 8.2%
woman (first author)/woman (last author), 12.0% man/
woman, 18.6% woman/man and 61.2% man/man. By race,
our references contain 14.8% author of colour (first)/author
of colour (last), 6.0% white author/author of colour, 18.0%
author of colour/white author, and 61.2% white author/
white author. Following the original method, a colour
author can be Asian, Hispanic, or Black. Of all 366 first/last
authors (counting duplicates), 73.2% are white, 22.4% are
Asian and only 4.4% are Hispanic and Black combined. This
method is limited in that the predictions made by gender-
api.com are not always correct. We have found that for a few
authors of Asian origins, gender-api.com predicted the incor-
rect gender while claiming to have a near-perfect confidence
score. We manually corrected a few predictions upon looking
up researchers’profiles, but the results cannot be assumed to
be completely accurate. The statistics we report fail to include
intersex, non-binary, transgender people, indigenous and
mixed-race authors, or those who may face differential
biases due to the ambiguous racialization or ethnicization of
their names. Under-representations of other dimensions such
as sexual orientation, (dis)ability, class and their intersections
are difficult to reveal. We look forward to future work that
could help us better understand how to support equitable
practices in science.
Data accessibility. This article has no additional data.
Declaration of AI use. We have not used AI-assisted technologies in creat-
ing this article.
Authors’contributions. H.K.: conceptualization, funding acquisition,
investigation, project administration, writing—original draft, writ-
ing—review and editing; G.L.: conceptualization, writing—review
and editing; R.N.: conceptualization, writing—review and editing.
All authors gave final approval for publication and agreed to be
held accountable for the work performed therein.
Conflict of interest declaration. We declare we have no competing interests.
Funding. H.K. is supported by the James S. McDonnell Foundation’s
Postdoctoral Fellowship for Understanding Dynamic & Multi-scale
Systems. We also acknowledge the support from ONR MURI grant
no. N00014-22-1-2616 to co-PIs R.N. and G.L.
Acknowledgements. We thank Dr Merihan Alhafnawi for providing feed-
back on the manuscript.
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