1. Physics of Living Systems

In-line swimming dynamics revealed by fish interacting with a robotic mechanism

  1. Robin Thandiackal  Is a corresponding author
  2. George Lauder  Is a corresponding author
  1. Harvard University, United States
https://doi.org/10.7554/eLife.81392
Share this article
Cite this article
  1. Robin Thandiackal
  2. George Lauder
(2023)
In-line swimming dynamics revealed by fish interacting with a robotic mechanism
eLife 12:e81392.
https://doi.org/10.7554/eLife.81392
  2. Download BibTeX
  3. Download .RIS

Abstract

Schooling in fish is linked to a number of factors such as increased foraging success, predator avoidance, and social interactions. In addition, a prevailing hypothesis is that swimming in groups provides energetic benefits through hydrodynamic interactions. Thrust wakes are frequently occurring flow structures in fish schools as they are shed behind swimming fish. Despite increased flow speeds in these wakes, recent modeling work has suggested that swimming directly in-line behind an individual may lead to increased efficiency. However, only limited data are available on live fish interacting with thrust wakes. Here we designed a controlled experiment in which brook trout, Salvelinus fontinalis, interact with thrust wakes generated by a robotic mechanism that produces a fish-like wake. We show that trout swim in thrust wakes, reduce their tail-beat frequencies, and synchronize with the robotic flapping mechanism. Our flow and pressure field analysis revealed that the trout are interacting with oncoming vortices and that they exhibit reduced pressure drag at the head compared to swimming in isolation. Together, these experiments suggest that trout swim energetically more efficiently in thrust wakes and support the hypothesis that swimming in the wake of one another is an advantageous strategy to save energy in a school.

Editor's evaluation

Why do fish school together? Energetic benefits have long been considered a key factor in motivating fish to swim together and tune their tailbeat to exploit the whirling wake generated by conspecifics. This study clearly demonstrates that fish benefit from swimming in a two-dimensional vortical wake by locating their body in the vortical low-pressure zones that passively impart a net thrust force on their oscillating bodies. The behavioural and biofluid mechanical findings will interest comparative biomechanists, movement ecologists, evolutionary biologists, fluid mechanists, and bioinspired roboticists.

https://doi.org/10.7554/eLife.81392.sa0

eLife digest

Some species of fish swim together in groups known as schools. This behaviour makes it easier to find food, avoid predators, and maintain social interactions. In addition, biologists also think that being in a group reduces the energy needed to swim compared to being alone.

Similar to the tracks that follow ships moving through water, fish also leave a wake behind them as they swim. By flapping their tail side-to-side, they create characteristic patterns in the water, including swirling currents. Fish in a school encounter many of these wakes from their neighbours, and may use this to position themselves relative to each other. Previous studies have suggested that swimming directly behind each other increases swimming efficiency; however, this was based on computer models and experiments on flapping systems rather than real-life settings.

To better understand how swimming in a line works in practice, Thandiackal and Lauder tested this idea in live fish. A robotic flapping foil designed to imitate the tail fin of a leading fish was placed in front of a single trout swimming in a tank with flowing water. The fish positioned itself directly behind the foil and timed its own flapping to match it. The trout also interacted with the swirling currents, which Thandiackal and Lauder calculated helped reduce the resistance from the water flow.

These results suggest that swimming directly behind each other can improve swimming efficiency, complementing previous studies showing the benefits of other formations, such as swimming side-by-side. This suggests that fish in schools may have many opportunities to save energy. In the future, this improved understanding could help to design underwater vehicles that work more efficiently in groups.

Introduction

Individuals in fish schools have long been hypothesized to benefit from hydrodynamic advantages associated with swimming near other conspecifics (Becker et al., 2015; Li et al., 2021; Park and Sung, 2018; Weihs, 1973). Recent work supports this hypothesis on the basis of experiments where schooling fish exhibit reduced tail-beat frequencies relative to solitary individuals which suggests decreased energy consumption by the group as a whole (Ashraf et al., 2017; Marras et al., 2015). A number of specific mechanisms have been proposed and investigated to show how corresponding hydrodynamic effects could contribute to reduced energy demands in schools (Figure 1). The phalanx or soldier formation describes fish swimming side-by-side, parallel to each other (Figure 1A), and fish in this position are expected to benefit from the channeling/wall effect and simulation studies Daghooghi and Borazjani, 2015; Hemelrijk et al., 2015 have shown increased efficiency for this formation. And Ashraf et al., 2017 linked the phalanx formation to reduced energy consumption in red nose tetras swimming in a school. Another beneficial interaction can occur when two fish swim in close proximity to one another (Figure 1B). Here, the leading swimmer is thought to experience increased thrust because of the additional effective added mass at the tail trailing edge due to blockage of water by the trailing swimmer behind. Simulations on pitching foils Bao and Tao, 2014; Saadat et al., 2021 have confirmed this effect and show increased overall hydrodynamic efficiency for the two-body system of leading and trailing swimmers. Measurements of reduced tail-beat frequencies of fish swimming at the front of schools of gray mullet compared to swimming in isolation further support these findings (Marras et al., 2015).

Figure 1
Download asset Open asset
Schooling positions with hydrodynamic benefits.

(A) Swimming side-by-side can increase thrust and efficiency by making use of the channeling effect (Ashraf et al., 2017; Daghooghi and Borazjani, 2015). (B) Leading swimmers benefit from higher thrust production due to increased effective added mass at their trailing edge stemming from the blockage of the water in close proximity to trailing swimmers (Bao and Tao, 2014; Saadat et al., 2021). (C) Trailing fish face reduced oncoming flows between two leading fish when swimming in a diamond formation (4). (D) Leading-edge suction provides propulsive thrust for a fish in a trailing position (Kurt and Moored, 2018; Maertens et al., 2017; Saadat et al., 2021).

A third commonly proposed schooling arrangement is the diamond or staggered pattern (Figure 1C) first suggested by Weihs, 1973. The value of swimming in this formation is due to the nature of thrust wake vortical structures generated behind swimming fish. Fish thrust wakes are characterized by both a vortex street of alternating orientation, and an increased average flow speed compared to the free stream (Blickhan et al., 1992; Müller et al., 1997; Nauen and Lauder, 2002; Tytell, 2010). Weihs hypothesized that fish directly behind another would experience a higher relative velocity and would have to exert extra energy and suggested that the most efficient swimming position lies midway between two preceding fish (Figure 1C) resulting in a diamond formation. A fish swimming in this diamond formation encounters flow conditions resembling a von Kármán drag wake, similar to the one shed by a cylinder under sufficiently high flow speeds. Liao et al., 2003 explored this scenario in trout and found reduced muscle activity for fish swimming in a drag wake, and direct measurements of energy consumption confirm that fish experience reduced energetic costs when in a drag wake (Taguchi and Liao, 2011).

In contrast to Weihs’ argument that in-line fish positions are disadvantageous (Figure 1D), some recent work suggests that swimming in tandem provides hydrodynamic advantages. Simulations (Hemelrijk et al., 2015; Maertens et al., 2017), flapping foil experiments (Boschitsch et al., 2014; Kurt and Moored, 2018), and robot experiments (Saadat et al., 2021) indicate increased thrust production and efficiency when a fish or flapping foil swims in a thrust wake. The fluid dynamic benefits to the follower occur because the swimmer in the thrust wake experiences the oncoming flow at its leading-edge with an oscillating angle of attack and is subject to lift forces that have components in forward direction. Maertens et. al (Maertens et al., 2017) argue that a downstream swimmer can reduce its drag by consistently turning its head in a manner that employs the oncoming vortex flow to increase the transverse velocity across the head. As a result, the pressure drag at the head can be decreased substantially and result in increased efficiency.

Although recent modeling work suggests advantages for in-line swimming, experimental data on live fish exploiting these conditions is lacking. Do live fish actually take positions in a thrust wake when free to swim at any location in flow? When fish swim directly behind another, do they alter their swimming kinematics and is there evidence for a reduction of swimming cost even when in a thrust wake with accelerated mean flow? Here we explore how fish interact with thrust wakes in a controlled experimental setting. We chose trout (brook trout, Salvelinus fontinalis) for our investigation as this species swims against oncoming currents in their natural habitat and is known to sense and take advantage of flow structures that can reduce energy use (McLaughlin and Noakes, 1998; Shuler et al., 1994). Fish moving in fluids use (1) vision, (2) the lateral line, and (3) the vestibular system to control their body motion. All of them have been the subject of numerous studies over the years (Ali, 2013; Coombs and Montgomery, 2014; Platt, 1973). The individuals in our experiments had all of these sensor modalities available. In our approach we emulate the thrust wakes from leading swimmers using an actuated flapping foil that serves as the artificial counterpart of a fish tail-fin. Similar approaches have been proposed in previous work to study attraction of fish to robots (Marras and Porfiri, 2012; Polverino et al., 2013) and how fish respond to thrust wakes (Harvey et al., 2022; Zhang et al., 2019). Using a flapping foil allowed us to generate accelerated flows with similar hydrodynamic characteristics, in terms of the Strouhal number and the relative axial and lateral spacing of shed vortices, to those of live fish (Anderson et al., 1998; Buchholz and Smits, 2005). By carefully choosing the robotic flapping motion, we generated fish-like thrust wakes and introduced trout to these conditions. We found that trout swim in-line with the flapping foil (Videos 1 and 2) and reduce their tail-beat frequencies compared to swimming at the same effective flow speeds under free-stream conditions. Further analyses employing particle image velocimetry revealed that individuals interact directly with oncoming thrust wake vortices. Finally, our pressure field computations showed reduced average pressures at the leading-edge suggesting reduced pressure drag and reduced swimming costs. These findings support the hypothesis that fish can reduce swimming costs under in-line swimming conditions and help explain why in-line swimming is common in schools of fish.

Video 1
Download asset
Trout swimming in the thrust wake of a flapping foil (bottom and side views).
Video 2
Download asset
Time lapse of a trout exploring the flow tank with a thrust wake present (bottom view).

Over time trout position themselves in the thrust wake and synchronize with the flapping foil.

Results

Reduced frequency and synchronization with a flapping foil

Artificial thrust wakes were generated in a recirculating flow tank using an actuated flapping foil with 2 degrees of freedom which enabled side-to-side movement as well as rotation (‘Materials and methods: Flapping foil,’ Figure 2). The motion of the foil together with the flow speed (St=0.267) were chosen such that the Strouhal number falls in the typical range of 0.2–0.4 for swimming fish (Saadat et al., 2017). The thrust wake generated by the flapping foil is characterized by a reverse Kármán vortex street and increased flow speeds in the wake (Figure 2—figure supplement 1) comparable to the wakes generated by swimming trout (Nauen and Lauder, 2002). Matching the Strouhal number of swimming fish and our flapping foil ensures similar hydrodynamics in terms of the relative axial and lateral spacing between vortices. It is worth noting that the relatively large span of the flapping foil induces thrust wakes along a larger depth and thus increases the chance that fish encounter the wake in the flow tank, however at the expense of producing two-dimensional (planar) thrust wakes.

Figure 2 with 1 supplement see all
Download asset Open asset
Experimental setup.

Flapping foil with 2 degrees of freedom (yaw and sway) generating a fish-like thrust wake in the flow tank. Trout swam in the dark while we captured the kinematics by means of high-speed cameras from a bottom and side view and using infrared lights for illumination. Low light in the tank upstream of the flapping foil allowed fish to orient. In separate experiments, we captured the flow dynamics using particle image velocimetry. We were able to record the entire flow field around the fish by using two lasers (in front and behind) simultaneously.

We used a paired experimental design and had the same individuals swim under two conditions: in a flow tank with (1) an actuated flapping foil generating a thrust wake, and (2) under control free stream conditions with the foil held in a stationary position in the water. In both conditions, the flow was fixed at the same speed, and thus permitted a controlled comparison of the corresponding swimming patterns. In addition, we carried out the same experiments 2.5 months apart, which allowed us to investigate how differences in body size affect the behavior under the different conditions as fish were larger in total length after this growth period. We captured the swimming kinematics using high-speed video recordings from the ventral perspective and extracted body midlines (‘Materials and methods: Experimental setup and Kinematic analysis’).

We found that trout from both size groups significantly reduced their tail-beat frequencies when they were exposed to thrust wakes (Figure 3A, Video 2). Smaller fish showed a decrease of 28.3%, and larger fish showed a decrease of 14.7% in the mean frequency. This suggests that fish maintained their position in the thrust wake by beating their tails less often than when they swam at the same ground speed in free-stream flow. These experiments further showed that fish were synchronizing their tail-beat frequency to that of the flapping foil. For both smaller (mean ± s.d.: 2.15 ± 0.29 Hz) and larger (1.96 ± 0.04 Hz) fish the swimming frequency approached the 2 Hz flapping foil motion when swimming in the foil thrust wake.

Figure 3 with 1 supplement see all
Download asset Open asset
Body kinematics in thrust wakes.

(A) Reduced tail-beat frequencies and (B) reduced overall phase lags for small (n = 4) and large (n = 6) trout swimming in the thrust wake compared to steady swimming at the same flow tank speed. (C) Illustration of the bending pattern by means of joint angles (rainbow colored lines) along the body. Black markers indicate the bending phase.

We analyzed body bending kinematics and identified decreased overall phase lags along the body in the thrust wake in both size groups (Figure 3B) compared to the free stream control condition. As a result, the bending of consecutive body segments was timed closer together (Figure 3C). Smaller overall phase lags also relate to fewer waves along the body. We did not find any significant differences in body amplitude between fish that swam in thrust wakes and in the free stream (Figure 3—figure supplement 1).

Reduced tail-beat frequencies towards the ones of the flapping foil and a change in body phase lags indicate that fish are synchronizing their movements to the flapping foil. To further investigate synchronization, we measured the phase difference between fish and the flapping foil as a function of the distance between them (Figure 4, Video 3). We found a linear relationship (R2=0.93) showing that the phase difference increases as fish are swimming further away from the foil. This result demonstrates that fish time their body undulations and tail-beats depending on their location in the thrust wake, and it suggests that they synchronize their movements to the oncoming vortices that are shed by the flapping foil.

Figure 4
Download asset Open asset
Phase difference between foil and fish.

(A) Linear relationship (n = 10) between phase difference and distance from the foil for fish swimming in-line in the thrust wake. Video 3 shows videos of the individual data points 1–10. (B) Distance is measured between the trailing edge of the foil (R1) and the leading edge of the fish (R2). The phase difference is measured between trailing edges of the foil (R1) and the fish (R3).

Video 3
Download asset
Analysis of phase difference between flapping foil and fish swimming in the thrust wake.

Panels 1-10 show the individual data points and illustrate how the phasing of the tail-beat changes linearly with the distance from the foil. Numbers on the panels correspond to the point numbers in the graph in the lower right, and to those in Figure 3.

How do fish interact with the flow in thrust wakes?

Kinematic analysis of fish swimming in thrust wakes indicates a frequency and phase synchronization with the flapping foil. To investigate flow dynamics and how the thrust wake generated by the foil interacts with the bending fish body we employed particle image velocimetry (‘Materials and methods: Setup to capture flow dynamics,’ Video 4) to visualize flow structures in the thrust wake during fish swimming trials. Analysis of flapper wake velocity fields show that trout in the thrust wakes interact with oncoming vortices that are shed from the flapping foil and time their movements accordingly. We identified two scenarios that we call double-sided and single-sided vortex interaction. Double-sided vortex interaction (Figure 5A–F, Video 5) is characterized by an initial vortex interception which splits the vortex in two parts. One part of the vortex stays attached and ‘rolls’' downstream along the body, whereas the other part is shed laterally and moves away from the body. In this situation, the trout body alternates between clockwise and counter-clockwise vortices that are intercepted, stay attached and roll along corresponding alternate sides of the body. Fish are thus able to ‘catch vortices on both sides of the body.

Video 4
Download asset
Trout swimming in a laser sheet used for particle image velocimetry (bottom and side views).
Figure 5
Download asset Open asset
Interactions between fish and vortices.

Two representative sequences over one swimming cycle with ventral view of trout station-holding in the thrust wake near the foil at distance d1 with double-sided vortex interactions (A–F) and located more downstream at d2 with single-sided vortex interactions (G–L). Oncoming vortices from the flapping foil are intercepted by trout in the wake. The vortices stay attached on one side depending on their orientation and ‘roll’ downstream along the body (velocity fields shown after subtraction of mean flow speed).

Video 5
Download asset
Visualization of vortex flow structures that interact with a trout swimming in the thrust wake (bottom view).

Single-sided vortex interactions (Figure 5G–L) undergo the same process of ‘catching vortices’; however, only for one of the two differently oriented vortex types that are shed from the foil. Consequently, the intercepted vortex stays attached and ‘rolls’ only along one side of the body. The single-sided vortex interactions are related to a slight lateral offset of fish position with respect to the center line around which the foil is oscillating, whereas double-sided vortex interactions occur for fish swimming directly in the center line. We also note that vortex interactions occurred at different distances with respect to the foil (d1 and d2 in Figure 5). The consistent interaction pattern between the trout body and oncoming vortices indicates that these fish are synchronizing their movements with respect to the flapping foil and the corresponding vortices shed into the wake.

Decreased head pressure indicates reduced energy requirements

A large part of the drag on a swimming fish at Reynolds numbers greater than 5000 is caused by drag forces at the anterior portion of the body that faces the oncoming flow (Du Clos et al., 2019; Lucas et al., 2020). Total body pressure drag on a swimming streamlined fish like trout is mainly determined by the pressure acting on the head (F=p⋅S, F: drag force, p: pressure, S: surface area). Therefore, to estimate the effect of swimming in a thrust wake on drag we compared pressure fields of fish swimming in the free stream to thrust wake conditions.

We derived the pressure fields at the anterior part of the fish body based on velocity field changes as proposed by Dabiri et al., 2014 (‘Materials and methods: Pressure field computation’). The computed pressure fields revealed reduced average head pressures in the thrust wake (Figure 6A and B, Figure 6—figure supplement 1). We found the strongest decreases (46% and 86% decrease compared to free-stream swimming) for fish swimming close to the foil and exploiting double-sided vortex interactions. Fish swimming further away from the foil and exhibiting single-sided vortex interactions also showed reduced average head pressure magnitudes compared to free-stream swimming (45% decrease). Here, we found an asymmetric average pressure pattern with higher average pressures at the side closer to the centerline of the foil oscillation. The other side of the head experienced smaller average pressures.

Figure 6 with 1 supplement see all
Download asset Open asset
Reduced head pressure in the thrust wake.

Average pressure fields of a trout swimming in free-stream flow (A) and in the thrust wake of a flapping foil (B) show reduced positive pressures (46% decrease) around the head despite increased oncoming flow. Consistent instantaneous positive pressures over time are present under free-stream flow conditions (C1–C4). Corresponding instantaneous pressure fields display alternating positive and negative pressures around the head in the thrust wake over time (D1–D4).

To understand how the average head pressures in the thrust wake were reduced despite faster oncoming flows caused by the flapping foil, we analyzed the instantaneous pressure fields (Figure 6D1–D4). Here, it becomes evident that the flapping foil induces oscillating negative and positive pressure zones around the head. The negative pressure (suction) zones cause forward thrust forces, whereas the positive pressures contribute to drag. On average, this reduces overall head drag as the positive pressure magnitudes are comparable to free-stream swimming but mean head pressure is reduced by the occurrence of negative head pressures for part of the cycle. This pressure analysis indicates that the drag of fish swimming in thrust wakes is reduced compared to free-stream swimming, and therefore supports the hypothesis of decreased energy used to hold station in thrust wakes with accelerated mean flow.

Discussion

In schools of swimming fishes, there are a number of different hydrodynamic effects that can be exploited to save energy by individual fish in various positions (Figure 1). Previous work has demonstrated benefits for swimming side-by-side (phalanx configuration), pushing off near followers, and forming diamond patterns (Ashraf et al., 2017; Saadat et al., 2021; Taguchi and Liao, 2011). But fish in schools often assume an in-line configuration with one fish swimming directly behind another (Video 6). The benefits, if any, of swimming in this tandem swimming mode have been the subject of some debate. Some authors (Verma et al., 2018; Weihs, 1973) suggest that swimming in tandem is not an energetically favorable configuration due to the accelerated wake flows generated by the fish in front. Other, primarily computational studies have suggested that a trailing streamlined shape could in fact experience reduced energetic cost due to leading edge suction resulting from an oscillating flow impinging on the head or leading edge of the trailing fish or foil (Kurt and Moored, 2018; Maertens et al., 2017; Saadat et al., 2021). To date, however, no experimental study has demonstrated that live fish will voluntarily swim in a thrust wake and that reduced swimming cost could result from such a position. With these experiments, we document that trout indeed perform volitional in-line swimming with their body located within the accelerated flow region, and our analysis suggests that they can save energy under these conditions.

Video 6
Download asset
Schooling silversides, Menidia menidia, swimming in a flow tank and exhibiting in-line swimming (bottom and side views).

Comparisons to drag wake swimming and differences from drafting

A drag wake in the context of fish swimming and schooling is characterized by a von Kármán vortex street between two thrust wakes (e.g., shed by two fish swimming parallel to each other). Drag wakes can be emulated behind cylinders when they are exposed to sufficiently high flow speeds. It is important to note that the average flow speed in a drag wake is inherently slower than in the free stream. This is highlighted in experiments that demonstrate a dead fish propelling itself forward in the wake of a cylinder (Beal et al., 2006). Intuitively, we can draw an analogy of a cyclist drafting behind another cyclist where the individual behind experiences reduced energy consumption while maintaining the same speed. This situation is an example of a drag wake and the reduced costs that ensue from moving in that reduced velocity zone: trailing cyclists benefit from the reduced relative oncoming flow which results in reduced aerodynamic drag.

The dynamics of thrust wakes, however, differ from drag wakes. Vortex orientations are reversed compared to the drag wake (termed a reverse Kármán vortex street), and, notably, thrust wakes are characterized by a higher average flow speed than in the free stream. For swimming fish, this is a consequence of tail fin movement which actively generates thrust so that individual fish located behind this thrust wake experiences higher than free stream mean flow velocities. Given this increased oncoming flow speed it is surprising that fish choose to swim in thrust wakes. If we return to our example of cyclists, it would correspond to a (fictional) case where a leading cyclist would have a propeller attached to their bicycle that generates additional thrust. The trailing cyclist would face an increased oncoming flow and experience increased aerodynamic drag. An in-line, tandem, formation might be expected to be disadvantageous in this case.

How does this situation differ from fish swimming in a thrust wake of a conspecific? A key difference is the undulatory characteristic of thrust wakes that are produced by swimming fish. A trailing fish faces the oncoming flow at its head with an oscillating angle of attack, and (unlike the trailing cyclist) the trailing fish oscillates its head during swimming (Di Santo et al., 2021) further enhancing the time-dependent variation in flow in the head region. Our analysis showed that as a result of the oscillatory wake impinging on the fish head the pressure distribution in the head region is composed of both positive and negative pressures, and thus effectively reduces the overall pressure drag. This is in agreement with previous simulation analyses (Maertens et al., 2017; Saadat et al., 2021) and makes in-line swimming an advantageous formation for fish.

Swimming efficiency in thrust wakes

In our experiments, we found that fish in thrust wakes significantly reduced their tail-beat frequency and the frequency, higher when swimming in the free stream, shifted toward the flapping foil frequency. We also found a linear relationship between phase difference and distance from the foil when fish swam in-line in the thrust wake, indicating that they are phase synchronized with the foil. Together with our PIV analysis, these results suggest that trout are exhibiting vortex phase matching (Li et al., 2020) which has been identified as an energy saving mechanism. These benefits are further supported by our computational analysis where we found lower pressure drag in fish swimming in the thrust wake. Apart from the reduced hydrodynamic resistance, the lower tail-beat frequencies without significant changes in amplitude also are reflective of reduced metabolic cost (Ohlberger et al., 2007; Steinhausen et al., 2005). The change in phase lags that we observed further suggests a change in the muscle activation pattern along the body. Decreased muscle activation was observed by Liao et al., 2003 when trout swim with a Karman gait in a drag wake, and passive dynamics could therefore be additional sources for energy savings that would however need to be confirmed in future experiments. Overall, our data suggests that fish can swim more efficiently in thrust wakes because they maintain the same swimming speed as in the free stream while facing reduced pressure drag and spending less energy by beating their tail less frequently.

These results are in agreement with past simulation and robot studies. (Harvey et al., 2022) adopted a similar approach to ours and exposed rainbow trout to a hydrofoil thrust wake. They also observed altered swimming gaits and estimated energy savings but via measurements of acceleration. It should also be noted that they used juveniles about three times smaller than trout used in our study, however at similar foil cord length. This indicates that fish in thrust wakes may exhibit energetic benefits over an extended size scale. (Verma et al., 2018) explored simulated leader-follower formations in a reinforcement learning framework. A first set of optimizations with a reward function based on a modified Froude efficiency led to formations in which followers settled close to the center line of the leader’s wake and showed well-coordinated behavior of the follower with the wake. In the same work, Verma et al., 2018 concluded in a second set of simulations that swimming in-line with a leader is not associated with energetic benefits for the follower. It is important to note that this conclusion was drawn based on a swimming strategy in which the follower would strictly try to attain an in-line position regardless of energetic considerations. As a consequence, the optimization led to an increased swimming amplitude, which permitted in-line swimming but at a higher energetic cost. We confirm that in-line swimming by itself is not necessarily energetically beneficial. More importantly, efficient swimming requires the correct timing of interactions with the wake. Rather than viewing in-line swimming as a policy, we can see in-line positions as favorable conditions to maintain wake synchronization as we found in our experiments of double-sided vortex interactions.

Finally, in-line swimming has been dismissed as a beneficial strategy in the past considering the diverging characteristics of three-dimensional compared versus two-dimensional wakes (Verma et al., 2018). In such cases, it could be argued that the area around the centerline of the wake is composed of quiescent flow and in-line swimming offers no opportunity to interact with vortices. Whereas these diverging wakes are predominantly found in simulation studies at lower Reynolds numbers (Borazjani and Sotiropoulos, 2008; Liu and Dong, 2016; Verma et al., 2018), we found no evidence for bifurcating wake structures behind trout swimming in the free stream, a finding in line with previous analyses of trout wake flow patterns (Müller et al., 1997; Tytell, 2010). The artificial thrust wake generated using our robotic flapping foil produced a parallel vortex street similar to our observations of the wake in freely swimming trout. In addition, given the experimental data on single-sided vortex interactions, we hypothesize that energy-efficient thrust wake interactions could also occur in diverging wakes but with a small offset to the centerline.

Limitations of this study

Previous studies have highlighted the three-dimensional (3D) effects of fish swimming. The 3D kinematics of the tail are determined by the 3D body shape as well as the motion (Tytell et al., 2008), and it has been shown that the tips of the caudal fin are subject to cupping into the flow. The flapping foil in our study is rigid and does therefore not exactly replicate this motion pattern. Future work could address this gap by using a flexible flapping foil. Another aspect of 3D fish swimming are 3D vortex rings that are shed into the thrust wake at the caudal fin. These structures induce flow in (1) lateral (side-to-side) and axial (forward-backward) swimming directions as well as in (2) the vertical (up-down) swimming direction. The flapping foil in our experiments spanned across an extended depth (‘Materials and methods,’ Figure 2) and generates 2D thrust wakes that produce the lateral and axial flow characteristics of vortex rings. These flow components are arguably important for thrust generation to swim forward and we showed in our study how fish interact with these flows and benefit from reduced pressure drag. Vertical components need to be included in future experiments to address how the up-and-down flow dynamics impact fish in a thrust wake. Nonetheless, our results contribute to a better understanding of in-line swimming in thrust wakes and are likely to extrapolate to 3D fish swimming.

In this study, we analyzed a limited number (n = 3) of swimming trials using PIV. Our goal was to use these trials to investigate the underlying mechanism of vortex interactions following the kinematic analysis that showed both frequency and phase synchronization between fish and the flapping foil. We were able to gain insight on single and double-sided vortex interactions and on reduced pressure drag via computational inference. We expect that there is a critical distance from the foil centerline at which fish transition from double to single-sided interactions. To further identify and quantify this critical distance a larger sample size is required. We will also require more swimming trials to quantify the reduction in pressure drag depending on the distance from the foil. This could help to understand if and where there are distances that are optimal in terms of energy savings, as, e.g., suggested in Saadat et al., 2021.

We know from past studies that there are a number of hydrodynamically beneficial schooling positions (Figure 1). Our results complement this body of work with regard to fish swimming in thrust wakes that are shed by leading individuals, a condition encountered within fish schools during in-line locomotion. In our controlled experiments, we show that trout volitionally swim in thrust wakes and exhibit advantageous flow interactions with incoming vortices suggesting increased energy efficiency for the in-line swimming condition. These results highlight the hydrodynamic complexity of fish schooling and support a view in which individuals in schools have a variety of opportunities to save energy when they swim side-by-side, in drag wakes, and in thrust wakes behind each other.

Materials and methods

Animals

We used brook trout, S. fontinalis, and carried out the same experiments 2.5 months apart to investigate size effects as the total lengths of the fish increased after this growth period. Smaller fish had a body length of BL = 15.8 ± 0.5 cm (n = 4, Re = 43,091, Re = u⋅BLν , u=0.3 ms, ν = 1.1⋅10-6m2s) and larger fish had a body length of BL = 19.3 ± 1.0 cm (n = 6, Re = 52,636). Particle image velocity trials were carried out for larger fish (n=3). Trout were held at a water temperature of 16°C and all experiments were performed in accordance with Harvard animal care and use guidelines, IACUC protocol number 20-03-3 to GL.

Experimental setup

Request a detailed protocol

We carried out all our experiments in a custom flow tank with flow speed control (Figure 2, Video 1). Fish were able to move freely in a space of 28 cm × 28 cm × 64 cm, where the front and back portions of the swimming section were limited by baffles. Given fish body widths of 3 cm or less, the side-to-side dimensional comparison of tank width to trout width has a ratio of about 10:1. Corrections for fish blocking of flow are not needed for less than 10% of the cross-sectional area (Kline et al., 2015). The boundary layer thickness in this tank is approximately 5 mm and has been quantified in Tytell and Lauder, 2004. We also only considered swimming trials in the center region of the tank for our analysis.

Flapping foil

Request a detailed protocol

We used a symmetric 3D printed NACA 0012 airfoil (cord: 67 mm; span: 190 mm; thickness: 8.1 mm; center of rotation: 48 mm from trailing edge; material: transparent photopolymer [RGD810] from an Connex 500 3D printer) as in previous studies of biomimetic propulsion and to emulate a fish body shape (Karbasian and Esfahani, 2017; Lauder et al., 2007; Shua et al., 2007; Van Buren et al., 2019; Zhang et al., 2019). It was actuated in sway and yaw direction (Figure 2) to mimic the tail-fin portion of swimming fish to induce fish-like thrust wakes. It is worth noting that we are using a rigid foil whereas the fish tail is flexible. Our goal was to reproduce similar tail tip excursions, therefore the corresponding movement was parametrized as follows:

(1) ysway=aswaysin(2πft),
(2) ϕyaw=ayawsin(2πft − π2)

Here, asway and ayaw denote the sway and yaw amplitudes, respectively. f indicates the flapping frequency, and t the time in seconds. The two motions are offset by a phase shift of π2 , which ensures that maximal yaw is reached whenever the foil crosses zero sway. For the purpose of our experiments asway=1 cm, ayaw=20° were selected and resulted in a peak-to-peak tail-beat amplitude of A=4 cm, which is comparable to the width of wakes in fish. Together with a frequency of f=2 Hz, this resulted in a Strouhal number of St = A⋅fU=0.267 and a Reynolds number of Re = ULν=20100 with U=0.3ms , L = 6.7 cm (cord length), and ν=10-6m2s , thus operating in a turbulent flow regime.

Use of NACA 0012 foil

Request a detailed protocol

A large body of work suggests that NACA 0012 foils are appropriate for the purpose of creating fish-like wakes (Anderson et al., 1998; Triantafyllou et al., 1993; Triantafyllou et al., 2004). They demonstrate that these foils show high propulsive efficiencies if they are operated in a Strouhal number range of 0.25–0.35 and that the reverse Karman vortex street with its increased wake velocities is related to thrust production. In part, high efficiency is associated with leading-edge vortices that are convected downstream and interact with trailing-edge vortices that result in a reverse Karman street (Anderson et al., 1998). Furthermore, previous studies have shown that fish operate in a similar range of Strouhal numbers and also produce reverse Karman vortex streets (Saadat et al., 2017). In contrast to these studies, classic low Reynolds number (Re < 100,000) airfoil literature suggests that NACA 0012, a supercritical airfoil, suffers from laminar separation bubbles and high drag when operated at lower Reynolds numbers. This could indicate that there are more efficient foils, e.g., cambered, flat or flexible foils, that could be used to generate the wakes as in our study. These foils are more suited for subcritical Reynolds numbers and future studies could explore differences and similarities by comparing sub- and supercritical foils in the context of creating fish-like wakes. However, as mentioned above, we were able to generate thrust producing wakes characterized by reverse Karman vortex streets with increased wake velocities that were sufficient for the purpose of our experiments as they provide the flow structures that fish face in wakes of other fish.

Wall effects

Request a detailed protocol

We have previously investigated wall effects (Quinn et al., 2014a) for the same flow tank and with the same robotic flapper used in this study. Using a six-axis force/torque sensor propulsion speed, forces, and efficiency were quantified at varying distances from the wall. The results of this analysis show that no significant wall effects are to be expected in our experimental setup of trout swimming behind a flapping foil. In brief, Quinn et al., 2014a studied the propulsion of relatively large flexible panels (150 mm span by 195 mm length) at varying distances from both the wall and bottom of the flow tank. The panels, when moved, came as close as 15 mm from the wall, which is closer than the trout studied here that were in the middle of the tank (~30 cm on a side and 1 meter long working area for filming). Plastic panels of three flexural stiffnesses were studied (Quinn et al., 2014a; table 1) and these panels were moved at the leading edge (in a manner similar to the foil motion used in this manuscript) to generate a propulsive wave. The flexural stiffness of the panels in Quinn et al., 2014a was specifically chosen to encompass the range of actual fish flexural stiffness which range from 10 to 3 to 10–6 (Shelton et al., 2014) and panel B (Quinn et al., 2014a; table 1) matches the flexural stiffness of trout bodies.

Given the non-dimensional distance d/a (d = mean distance from the wall, and a = heave amplitude of the flexible body at the leading edge), Quinn et al., 2014a show that there is little to no effect on propulsive speed and economy for all panels at d/a > 5. For the most trout-like panel B, there is almost no effect at any distance. In this article, trout operated at a value of d/a between 7 and 3.5 depending on whether ‘a’ is taken as head oscillation or tail oscillation amplitude.

One additional study from Quinn et al., 2014b used another flapper/flow tank system to investigate the wall effect for a rigid pitching foil at different distances from the wall. Their data show that for conditions similar to that of our trout experiments where trout are 15 cm from the wall, there is no effect on propulsive forces.

Although swimming close to a surface with an undulatory body can certainly improve propulsive efficiency and alter the time-dependent profiles of forces and torques, these previous experiments using the exact same experimental system show that it is highly unlikely that wall effects have influenced our results for trout swimming in the center of the flow tank.

Fish kinematics

Request a detailed protocol

High-speed video was used to capture kinematic variables such as tail-beat frequency, body amplitudes, and phase lags during swimming trials. The experiments were carried out in the dark to provide a controlled environment with minimal external distractions. To provide the fish with some sense of visual orientation, a fiber light was installed upstream behind the front baffle in the flow tank. We then used infrared lights (Figure 2), which are outside the visual spectrum of trout, to provide the necessary illumination to capture the scene with high-speed cameras. We took video recordings at 125 frames per second from a ventral and an angled side view.

Particle image velocimetry

Request a detailed protocol

To capture the flow patterns during swimming trials we used particle image velocimetry (PIV) as in our previous work (Domel et al., 2018; Thandiackal et al., 2021b; Zhu et al., 2019). For this purpose, we seeded the water in the flow tank with near-neutrally buoyant plastic particles (~50 μm mean diameter) and used two lasers to create a light sheet around the swimming fish (Figure 2). Movements were then recorded at 1000 frames per second from a ventral and angled side view (Video 4). We used the side view to identify the location of fish with respect to the light sheet. Only swimming sequences where the laser light sheet passed through the middle of the swimming fish body were considered in our analysis.

Kinematic analysis

Body midlines

Request a detailed protocol

We used a custom MATLAB script to manually track 9 points along the body midline within a given frame. Piecewise cubic spline interpolation was then applied to generate smooth midline curves. We manually tracked the midlines in every sixth frame and linearly interpolated between these frames to obtain midline estimates for all frames that were recorded at 125 frames per second.

Frequency estimation

Request a detailed protocol

Tail-beat frequencies were determined by averaging the period between maximal lateral tail tip excursions over three consecutive swimming cycles for each swimming trial.

Phase lag estimation

Request a detailed protocol

The phase lag describes how the traveling wave of body bending propagates along the body. To quantify the body bending, we divided the body first into N=20 equal length segments and computed the joint angles ϕi between segments. The intersegmental phase lag Δϕi was then computed as the time delay between joint angles of consecutive segments as a fraction of the cycle duration T. As in Thandiackal et al., 2021a, we used cross-correlation of the joint angle signals to determine this time delay. Finally, we obtained an estimate of the overall phase lag ΔΦ by summing up the intersegmental phase lags along the entire body.

(3) Δϕi(t)=crosscor(ϕi , ϕi+1)T⋅2π,
(4) ΔΦ(t)=∑i=1NΔϕi(t)

We note that the wavelength λ=2πΔΦ (as a fraction of the body length) can be computed from the phase lag, and we report this metric in the supplementary data (Thandiackal and Lauder, 2022).

Body amplitude estimation

Request a detailed protocol

We define the amplitude along the body as the maximal displacement perpendicular to the forward direction of movement. Forward and lateral direction are determined by applying a principal component analysis on the point cloud of all tracked midline points from a given swimming trial. The first principle component (PC) that captures most of the variation represents the forward direction whereas the second PC represents the lateral direction. Based on these directions the lateral displacement at a given midline point in time is then determined as the projection of that point on the lateral direction. Finally, we determine the body amplitude as half of the range of lateral displacements at a given midline point over the duration of the swimming trial.

Phase difference estimation

Request a detailed protocol

To estimate the phase difference between the flapping foil and fish swimming in the thrust wake, we compared the lateral displacement of the trailing edges of the foil and fish. As for the body phase lag (see above), we used a cross-correlation of these two signals to determine the time delay as a fraction of the cycle duration.

Statistical analysis

Request a detailed protocol

To confirm hypothesized decreases in mean frequency and phase lag between the free stream control condition and swimming in the thrust wake, we carried out paired, one-sided Welch t-tests (assuming unequal variance). Significant differences in mean amplitude under these conditions were investigated using paired, two-sided Welch t-tests. p-Values are reported in Figures 2 and 3.

Pressure field computation

Request a detailed protocol

Pressure fields were inferred from particle image velocimetry (see ‘Experimental setup’). We followed the methodology described in our previous work (Thandiackal and Lauder, 2020). In brief, the pressure at each grid point is computed by taking the median over eight families of integration paths that each integrate the pressure gradients. The pressure gradients themselves are estimated based on sequential velocity fields and zero pressures are assumed at the domain boundary where paths are initiated. Velocity fields were computed in DaVis 8.3 (LaVision Inc) and the pressure fields were obtained using the Queen 2.0 software by Dabiri et al., 2014. Corresponding fluid–solid interfaces, that block integration paths, included both the flapping foil as well as the fish body and were extracted using custom MATLAB and Python scripts. This approach has been used and validated in previous publications (Lucas et al., 2017; Lucas et al., 2020; Thandiackal et al., 2021b).

We quantified the head pressures by averaging over a rectangular zone that extends from the fish snout and that spans 10% of the body length in axial direction and the width of the fish in lateral direction. Albeit arbitrarily defined, this allowed us to directly compare pressure drags for steady swimming vs. thrust wake swimming.

Data availability

Data that support the findings of this study are available on: https://doi.org/10.6084/m9.figshare.c.6093405; https://gitlab.com/robintha/052-matlab-midline-annotation (copy archived at swh:1:rev:f3e2d4b0afb18419ac816b7df45f5ad0630de59c); https://github.com/basilisklizard/fluid-structure-segmentation (copy archived at swh:1:rev:7383c1d1627b2285b6a29074479f78220e3bd502).

The following data sets were generated
    1. Thandiackal R
    2. Lauder GV
    (2022) figshare
    Data and Movies - In-line swimming dynamics revealed by fish interacting with a robotic mechanism.
    https://doi.org/10.6084/m9.figshare.c.6093405

References

  26. Conference
    1. Liu G
    2. Dong H
    (2016) Effects of Tail Geometries on the Performance and Wake Pattern in Flapping Propulsion
    ASME 2016 Fluids Engineering Division Summer Meeting collocated with the ASME 2016 Heat Transfer Summer Conference and the ASME 2016 14th International Conference on Nanochannels, Microchannels, and Minichannels.
    https://doi.org/10.1115/FEDSM2016-7691
  56. Book
    1. Tytell ED
    (2010) Do trout swim better than eels? challenges for estimating performance based on the wake of self-propelled bodies
    In: Taylor GK, Triantafyllou MS, Tropea C, editors. Animal Locomotion. Berlin Heidelberg: Springer. pp. 63–74.
    https://doi.org/10.1007/s00348-007-0343-x

Decision letter

  1. David Lentink
    Reviewing Editor; University of Groningen, Netherlands
  2. Christian Rutz
    Senior Editor; University of St Andrews, United Kingdom
  3. Maurizio Porfiri
    Reviewer; New York University, United States

Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.

Decision letter after peer review:

Thank you for submitting your article "In-line swimming dynamics revealed by fish interacting with a robotic mechanism" for consideration by eLife.

Your article has been reviewed by two peer reviewers, and the evaluation has been overseen by David Lentink as the Reviewing Editor and Christian Rutz as the Senior Editor. The following individual involved in the review of your submission has agreed to reveal their identity: Maurizio Porfiri (Reviewer #2).

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this decision letter to help you prepare a revised submission.

Essential revisions:

1) In the summary of potential energy-saving mechanisms in schooling fish, please include the "vortex-phase matching" mechanism (ref. 33 in the references); it is related to the mechanism found in this paper, as discussed in the Discussion section (Line 291).

2) The justification for the effectiveness of the rigid NACA airfoil is not clear. In Line 108, you mentioned "similar hydrodynamic characteristics", but what are these characteristics? In Line 125 and the Materials and methods, you mentioned similar reverse Karman vortex streets and increased speeds in the wake. A quantitative comparison would help justify and quantify the similarity. Further, we are missing a discussion of the Reynolds number regime. NACA airfoils are super-critical airfoils, meaning that they have been designed for high Reynolds numbers during which laminar flow separation is a minor issue. Operating such super-critical airfoils at sub-critical Reynolds numbers usually results in large-scale flow separation, such as laminar separation bubbles, poor stall behavior, and a drag crisis. Variable camber and a 3D shape (a fish-like body) or sharp leading edge of a 2D airfoil (optimal sub-critical airfoil shape) can dramatically reduce the issues. This is well established in the low Reynolds number aerodynamics literature and applies 1:1 to hydrodynamics, but is often overlooked by biomechanists working on swimming. Please outline the issues as limitations of the study in the methods and Discussion sections.

3) The novelty of the robotics-based experimental approach is overstated, given that several earlier studies have investigated the response of live fish to thrust wakes, generated by pitching airfoils or robotic fish. Please give due credit to previous studies and temper claims accordingly. Such strong novelty claims are not needed to make a compelling case for publication in eLife -- our policy is to focus on informative scientific contributions in the context of the literature.

4) It seems like the length of the test section in the experiments is on par with the body length of the animals, hence wall-effects could confound the study to some degree. Please use the established literature and any estimation within your reach to outline wall-effects in the methods, and discuss the results in light of this limitation. Provide guidance for follow-up studies on how they could improve on the current experimental design.

5) The role of 3D effects needs to be elaborated; the 2D experimental data limit the generality and conclusions on actual fish collective behavior. Please include these issues in the introduction, explain how you converged on this experimental design in the methods, and discuss the limitations in the discussion and how future studies could expand the current work to the full 3D realm. In the discussion, please cite papers comparing 2D and 3D wakes to infer any possible implications for interpreting the quantitative measurements of vorticity and pressure in the paper, and extrapolating these findings to 3D fish swimming close together. The merit of 2D studies is not questioned, but does limit the conclusions and should inspire future 3D studies to further test your conclusions and extrapolate the implications for actual (3D) fish collective behavior.

6) Please briefly comment on other sensory modalities (touch, the vestibular system, vision), citing the relevant literature, before going in-depth on which senses play a role, so general readers (across biology and engineering) can understand your framework and its justification/limitations. Expanding the introduction and discussion on sensory modalities beyond the chosen/motivated focus will help readers understand your scientific reasoning.

7) In the methods section, please provide detailed reasoning for the selection of particle image velocimetry plane chosen for the computation of vorticity from 2D data and inference of pressure from 2D data. Although the cited paper from Dabiri is helpful, a summary of the approach for pressure inference is essential to clarify its reliability. This knowledge cannot be assumed to be generally known among the readership.

8) Please clarify if the flow is laminar or turbulent in the tested flow regime and add a plot showing at what distance the switch between double vortex split and single-sided vortex occurs.

9) Please carefully double-check the accuracy of wording in "increase in flow speed in the thrust wake" in the abstract and introduction, which may be in contrast with flow refuging behind bluff bodies.

10) In Figure S2, it appears that the fish and airfoil differ, yet they appear to shed similar vortex pairs -- is this the case? If so, please discuss this matter in your experimental design (and discussion where relevant).

11) Line 158: The "one-sided" versus "two-sided" vortex interaction analyses are qualitative. Is it possible to quantitatively analyze the relationship between the phase of the airfoil and the phase of head movements, since the thrust wake is linked to the phase of the airfoil? In Line 184, you mentioned fish adopt two-sided or one-sided interactions depending on the lateral offset and the distance between the airfoil and real fish. Is it possible to estimate the critical distances which drive the fish to switch from two-side vortex interaction to one-side?

12) Since fish might also swim further away (e.g., S Movie 2), what are the parameters you used to define/filter in-line swimming patterns? Line 165: You state "time their movements accordingly" to interact with the wakes generated by the airfoil -- but is it possible to add quantitative analyses of how the fish may time their movements? For example, by determining the phase difference between the airfoil motion and fish head oscillation? This seems also applicable to Line 142. Finally, in Line 146, please explain to the general reader why you did not consider using fish body wavelength for determining phase lag, which is more typical for fish kinematics. If this is not possible, please justify your processing choices accordingly, so the general reader can understand your choices and any limitations this may have for interpreting the findings.

13) Line 122 states you chose St around 0.267. Please outline how you selected the oscillation frequency of the airfoil with respect to the natural tail beat frequency. Also, please clarify how you selected the experiment parameters a_{sway}, a_{yaw}, amplitude, since there are different combinations of these parameters possible that give St ~ 0.267, and not all combinations are biologically representative or informative. Please include typical kinematics data of real fish here to help justify the parameter selection and, whenever there is a clear deviation, please list it as a limitation in the introduction and include it in the discussion.

Line 35: Please clarify: "however, no data are available on live fish interacting with thrust wakes". Most of the previous studies (such as ref. 5 and 6) with real fish systems provide some data on live fish interacting with thrust wake such as kinematic data representative for the hydrodynamic interactions.

Line 129: "with the foil in a stationary position", this needs to be clarified for a general reader. Should we interpret this as the foil being in a stationary position above the water? Otherwise, should we assume it is generating a von Karman drag wake with vortices? Since these airfoils have large-scale flow separation (laminar separation bubbles) at these sub-critical Reynolds numbers (below roughly Re = 100,000 / 200,000 depending on airfoil shape). Please clarify in the methods (and discussion in case there may be interference).

Figure 2: Should Panel C not be A, since it illustrates the phase lag calculation? Please resolve.

Figure 3: Shouldn't the legend for the color bar be Vorticity along the z-axis?

Line 202: "46% and 86% decrease compared to free-stream swimming" -- does this mean there are only two cases analyzed here? How did you establish these values? What is the zone around the fish head you used to evaluate pressure to make the comparison? Does this limit the analyses? Please clarify in a way the general reader can understand.

Line 323: You claim "swimming in-line is a frequently occurring situation" -- how often do fish swim in line compared to other formations?

[Editors' note: further revisions were suggested prior to acceptance, as described below.]

Thank you for resubmitting your article entitled "In-line swimming dynamics revealed by fish interacting with a robotic mechanism" for further consideration by eLife. Your revised article has been evaluated by a Reviewing Editor, who has drafted the below feedback to guide final revisions. A Senior Editor has overseen the re-evaluation.

The article has been improved but there are some remaining issues that need to be addressed:

The authors need to fully address the helpful feedback from the reviewers on the limitations of their hydrodynamic analysis. NACA 0012 is not an appropriate Low-Reynolds number airfoil and suffers from laminar separation bubbles at low Reynolds numbers, a low Reynolds number drag-crisis resulting in high drag, and a stall hysteresis loop. Classic low Reynolds number airfoil literature on these issues should not be ignored. That does not make the work unpublishable, but the reader needs to know about these limitations regardless of the fact that indeed many previous papers ignored them -- so that future studies can also consider more appropriate airfoils to see if that makes a difference to conclusions. Demonstrating that this has indeed an insignificant effect in the present research paradigm would require comparing a sub- and supercritical airfoil and showing that the outcome is highly similar; especially considering the lift-drag ratio of NACA 0012 is so much lower than even a flat or cambered plate at a low Reynolds number.

Also, while wall effects may be acceptable for answering the present research question, given how close the walls are, there needs to be a more detailed discussion of possible limitations. Wall interference is characterized based on airfoil/ propulsive body length, not width. For largely separated flow phenomena, such as shed vortices, there can be a marked wall effect when they are this close. It may not invalidate the work, but the cited literature ("In addition, wall effects (Webb, 1993) have minor to no measurable effects on fish swimming in our experiments.") lacks the fluid mechanic depth to argue that there is no measurable effect. That said, the experiment does seem representative of fish swimming close to a solid boundary, and for those conditions, the findings appear robust. There is no reason to believe that the non-quantified wall effect would disqualify the work for understanding how fish swim further away from a wall. But eLife offers the space to report that wall effects were not quantified (in the way a fluid mechanist would expect this to be done; boundary layer thickness at the wall is not actually sufficient for this). Given how close the walls are in body lengths this requires more consideration in future work, for example, with a robot fish mounted on a force sensor. The authors do not need to do this work for inclusion in the present article, if claims are tempered appropriately.

Addressing the above points should not be much work, but will improve the rigor of future fish swimming research. The Reviewing Editor has kindly offered to answer any questions you may have about these revision requests.

https://doi.org/10.7554/eLife.81392.sa1

Author response

Essential revisions:

1) In the summary of potential energy-saving mechanisms in schooling fish, please include the "vortex-phase matching" mechanism (ref. 33 in the references); it is related to the mechanism found in this paper, as discussed in the Discussion section (Line 291).

We thank the reviewers for this suggestion, and we have added a new analysis, figure, supplemental movie, and discussion of this important point. In our revision we now included an analysis of the phase difference between the flapping hydrofoil and the fish swimming in its wake (Line 186-198). We found a linear relationship between the fish-foil distance and the phase difference (see new Figure 3 / supplemental video 3). This supports previous findings using robotic systems that have studied vortex-phase matching, and demonstrates that live fishes can vortex phase-match. We believe that this is the first demonstration of this phenomenon.

As suggested, we accordingly mention the vortex-phase matching mechanism (Li et al., 2020) in the first paragraph (L327-330) of the section on “Swimming efficiency in thrust wakes” (discussion). We wish to note, however, that in the Li et al. paper, the vortex phase matching occurs in fish swimming nearly side-by side, and not in the in-line configuration, so our experiments here address another aspect of vortex phase matching where fish swim in a tandem arrangement.

2) The justification for the effectiveness of the rigid NACA airfoil is not clear. In Line 108, you mentioned "similar hydrodynamic characteristics", but what are these characteristics? In Line 125 and the Materials and methods, you mentioned similar reverse Karman vortex streets and increased speeds in the wake. A quantitative comparison would help justify and quantify the similarity. Further, we are missing a discussion of the Reynolds number regime. NACA airfoils are super-critical airfoils, meaning that they have been designed for high Reynolds numbers during which laminar flow separation is a minor issue. Operating such super-critical airfoils at sub-critical Reynolds numbers usually results in large-scale flow separation, such as laminar separation bubbles, poor stall behavior, and a drag crisis. Variable camber and a 3D shape (a fish-like body) or sharp leading edge of a 2D airfoil (optimal sub-critical airfoil shape) can dramatically reduce the issues. This is well established in the low Reynolds number aerodynamics literature and applies 1:1 to hydrodynamics, but is often overlooked by biomechanists working on swimming. Please outline the issues as limitations of the study in the methods and Discussion sections.

Thank you for pointing out where we were unclear on the choice of the NACA 0012 hydrofoil and the missing discussion of the Reynolds number regime. We would like to emphasize that the NACA 0012 shape specifically is commonly used to generate fish-like wakes in studies of biomimetic propulsion and to emulate a fish body shape (Akhtar et al., 2007; Karbasian and Esfahani, 2017; Lauder et al., 2007, 2011; Shua et al., 2007; Van Buren et al., 2019; Zhang et al., 2019). Furthermore we revised the manuscript text (L121-122, L141-142) to better explain what we mean by “similar hydrodynamic characteristics”. As shown in previous work, fish wakes are characterized by their reverse Karman vortex street and increased average flow speeds in the wake. In our work, we focused on the geometry of the wake, i.e., similar relative axial and lateral spacing between vortices while maintaining a similar width wake in lateral direction (≈4cm). These properties are well defined via the Strouhal number and the width of the wake. We now report both Strouhal numbers of fish and foil as well as the wake widths (L460-462). In addition, we believe that our visualization of the wake in Figure S2 supports our claim of similarity. For completeness we also report the Reynolds number for the flapping foil as Re = ULν=20100, U=0.3ms, L = 6.7cm (cord length), ν=10−6m2s (L463-464). Given these data, we are confident that the foil is operating as expected and the issues related to laminar separation bubbles, poor stall behavior and drag crisis can be excluded.

3) The novelty of the robotics-based experimental approach is overstated, given that several earlier studies have investigated the response of live fish to thrust wakes, generated by pitching airfoils or robotic fish. Please give due credit to previous studies and temper claims accordingly. Such strong novelty claims are not needed to make a compelling case for publication in eLife -- our policy is to focus on informative scientific contributions in the context of the literature.

We appreciate the reviewer’s comment and have toned down the novelty of the robotics-based approach by including the following papers (L118-120):

  • Fish in the thrust wake of a hydrofoil (Harvey et al., 2022; Zhang et al., 2019)

  • Fish in the thrust wake of robotic fish model (Marras and Porfiri, 2012)

However, we do wish to add that there are very few papers on live fish swimming in a thrust wake, and almost all papers focus on fish in a drag wake in which energy savings would be expected. Here we show that energy savings can occur when fish swim in a thrust wake using pressure computations on the surface of freely-swimming fish, an approach that has not been previously undertaken.

4) It seems like the length of the test section in the experiments is on par with the body length of the animals, hence wall-effects could confound the study to some degree. Please use the established literature and any estimation within your reach to outline wall-effects in the methods, and discuss the results in light of this limitation. Provide guidance for follow-up studies on how they could improve on the current experimental design.

We have added text to the methods section in the manuscript to explain our view on wall effects in these experiments (which we believe to be minimal) (L437-441). The recirculating flow tank and experimental apparatus used for the experiments described in this manuscript have been used for more than 50 previous papers on fish locomotion some of which also use the flapping foil propulsion system to generate wakes. Below we provide some key citations and reviews to this previous work, and here we respond directly to the comment on fish size relative to flow tank dimensions. The test section is 30 cm square in cross section and over a meter long but is open at both the upstream and downstream ends to allow flow to enter and leave so blocking effects in this dimension are not an issue. The overall length of the tank in the longitudinal dimension is two meters. Trout in our experiments varied in both length from 15 to 20 cm, and were 3 cm or less in body width. Thus, the side-to-side dimensional comparison of tank width to trout width has a ratio of 10:1. The relevant literature on this topic (see below) indicates that corrections for fish blocking of flow are not needed for less than 10%, although these studies have not quantified flow patterns. But we have previously done this by using PIV to assess flow across the entire width of the flow tank, and did not observe any measurable effect of fish on flow in the far field even with the tail beating from side to side as vortices are rapidly convected downstream.

Also, since we have previously quantified the boundary layer thickness in this same tank (Tytell and Lauder, 2004) which was approximately 5 mm at the flow speeds of these experiments, and fish swam in the center of the tank, we do not believe that wall effects have any measurable effect on our results.

Some key references:

Tytell, E. D. and Lauder, G. V. (2004). The hydrodynamics of eel swimming. I. Wake structure. Journal of Experimental Biology 207, 1825-1841.

Webb, P. W. (1993). The effect of solid and porous channel walls on steady swimming of steelhead trout Oncorhynchus mykiss. Journal of Experimental Biology 178, 97-108.

Kline, R. J., Parkyn, D. C. and Murie, D. J. (2015). Empirical modeling of solid-blocking effect in a Blazka respirometer for gag, a large demersal reef fish. Adv. Zool. Bot 3, 193-202.

Lauder, G. V. (2006). Locomotion. In The Physiology of Fishes, Third Edition, (D. H. Evans and J. B. Claiborne, eds.), pp. 3-46. Boca Raton: CRC Press.

Lauder, G. V. (2015). Fish locomotion: recent advances and new directions. Annual review of Marine Science 7, 521-545.

Lauder, G. V. and Tytell, E. D. (2006). Hydrodynamics of undulatory propulsion. In Fish Biomechanics. Volume 23 in Fish Physiology, (R. E. Shadwick and G. V. Lauder, eds.), pp. 425-468. San Diego: Academic Press.

5) The role of 3D effects needs to be elaborated; the 2D experimental data limit the generality and conclusions on actual fish collective behavior. Please include these issues in the introduction, explain how you converged on this experimental design in the methods, and discuss the limitations in the discussion and how future studies could expand the current work to the full 3D realm. In the discussion, please cite papers comparing 2D and 3D wakes to infer any possible implications for interpreting the quantitative measurements of vorticity and pressure in the paper, and extrapolating these findings to 3D fish swimming close together. The merit of 2D studies is not questioned, but does limit the conclusions and should inspire future 3D studies to further test your conclusions and extrapolate the implications for actual (3D) fish collective behavior.

Thank you for your comment. We followed your suggestion and explicitly mention in the first paragraph of the results (L142-145) that we are employing a 2D experimental approach, and we discuss the limitations of our 2D experimental approach with respect to 3D effects in a dedicated paragraph in the discussion (L382-398).

In addition, we would like to emphasize in experiments such as these, in which fish swim in-line with a robotic flapping foil, that limiting the height of the foil (to effectively make the experiments more 3D) is quite challenging as fish are much less likely to voluntarily hold position due to the smaller foil dimensions in our relatively large experimental space. We do agree that follow-up experiments with a shorter, more 2D foil, would be valuable indeed, but believe that the data we present with a longer foil that makes live fish data acquisition more feasible, is a good first step.

6) Please briefly comment on other sensory modalities (touch, the vestibular system, vision), citing the relevant literature, before going in-depth on which senses play a role, so general readers (across biology and engineering) can understand your framework and its justification/limitations. Expanding the introduction and discussion on sensory modalities beyond the chosen/motivated focus will help readers understand your scientific reasoning.

Thank you for this suggestion. We added new text mentioning sensory modalities in the introduction (L112-115).

Fish moving in fluids use (1) vision, (2) the lateral line, and (3) the vestibular system to control their body motion. All of these modalities have been the subject of numerous studies over the years. It’s a lot to add a discussion of the various modalities and what they are used for to this manuscript, but we did add (L115-116) information to the effect that fish in our experiments had all modalities available to them for flow detection, a very basic description of the three modalities involved, and added some key references to each of the modalities as indicated below here:

Coombs, S. and Montgomery, J. (2014). The Role of Flow and the Lateral Line in the Multisensory Guidance of Orienting Behaviors. In Flow Sensing in Air and Water, pp. 65-101: Springer.

Platt, C. (1973). Central Control of Postural Orientation in Flatfish II. Optic-Vestibular Efferent Modification of Gravistatic Input. Journal of Experimental Biology 59, 523-541.

Ali, M. (2013). Vision in fishes: New approaches in research: Springer Science and Business Media.

7) In the methods section, please provide detailed reasoning for the selection of particle image velocimetry plane chosen for the computation of vorticity from 2D data and inference of pressure from 2D data. Although the cited paper from Dabiri is helpful, a summary of the approach for pressure inference is essential to clarify its reliability. This knowledge cannot be assumed to be generally known among the readership.

Thank you for this comment. We added a summary of the approach to compute pressure fields from 2D velocity fields (L536-542). In brief, we used the standard fish-PIV horizontal plane (common to many investigations over recent decades) as a basis for estimating pressures and forces, and we have used and validated this approach (against measured forces generated by a flapping foil) in previous publications (cited below, L544-545), and added these citations at this location to assist readers.

Thandiackal, R. and Lauder, G. V. (2020). How zebrafish turn: analysis of pressure force dynamics and mechanical work. The Journal of Experimental Biology 223, jeb223230.

Thandiackal, R., White, C. H., Bart-Smith, H. and Lauder, G. V. (2021). Tuna robotics: hydrodynamics of rapid linear accelerations. Proceedings of the Royal Society B: Biological Sciences 288, 20202726.

Lucas, K. N., Dabiri, J. O. and Lauder, G. V. (2017). A pressure-based force and torque prediction technique for the study of fish-like swimming. PLoS ONE 12, e0189225.

Lucas, K. N., Lauder, G. V. and Tytell, E. D. (2020). Airfoil-like mechanics generate thrust on the anterior body of swimming fishes. Proceedings of the National Academy of Sciences, 201919055.

8) Please clarify if the flow is laminar or turbulent in the tested flow regime and add a plot showing at what distance the switch between double vortex split and single-sided vortex occurs.

We now report the Reynolds number (Re=20100) in the ‘Flapping foil’ section in the Materials and methods (L463-464) and clarify that we are operating in a turbulent flow regime (Please also see our response to comment 2). And please see our response to comment 11 regarding the critical distance for double-sided vs single-sided vortex interactions.

9) Please carefully double-check the accuracy of wording in "increase in flow speed in the thrust wake" in the abstract and introduction, which may be in contrast with flow refuging behind bluff bodies.

We confirm the accuracy of the wording. Flow refuging is related to drag wakes, e.g., behind bluff bodies whereas we observe increased average flow speeds in thrust wakes. In this case, we have *accelerated* the flow beyond the free-stream using our robotic flapper that generates a fish-like reverse Karman vortex street.

10) In Figure S2, it appears that the fish and airfoil differ, yet they appear to shed similar vortex pairs -- is this the case? If so, please discuss this matter in your experimental design (and discussion where relevant).

We apologize for the confusion here, but the airfoil and the fish do indeed generate similar wakes! This has actually been used in previous studies that use flapping foils as models for fish-like propulsion. The flapping foil serves as a model for the tail fin portion of the fish body. Given that the fish body and its tail are flexible they have different kinematics than the rigid flapping foil. Nonetheless, by purposefully defining the kinematics of the rigid flapping foil we can achieve similar excursions of the tail tip and ultimately generate similar vortex pairs. We added more explanations in the ‘Flapping foil’ section in the methods (L446-451) as well as in the caption of Figure S2. Please also see our response to comment 13. Also, the use of a flapping foil model system to generate fish-like wakes is quite established in the fish bio-fluids literature (both computational and experimental) and we cite several relevant papers in the revised manuscript.

11) Line 158: The "one-sided" versus "two-sided" vortex interaction analyses are qualitative. Is it possible to quantitatively analyze the relationship between the phase of the airfoil and the phase of head movements, since the thrust wake is linked to the phase of the airfoil? In Line 184, you mentioned fish adopt two-sided or one-sided interactions depending on the lateral offset and the distance between the airfoil and real fish. Is it possible to estimate the critical distances which drive the fish to switch from two-side vortex interaction to one-side?

Many thanks to the reviewers for their suggestions to further investigate phase relationships, as we believe that this has significantly improved the manuscript. As mentioned in our response to comment 1, we have added a quantitative analysis (L186-198) of the phase difference between the hydrofoil and fish swimming in its wake (see the new figure and the new supplemental movie). As a result, we found that the phase difference varies linearly with the distance between fish and foil. We agree that the analysis of double-sided vs single-sided vortex interactions can be considered mostly qualitative at this point, although it is based on the quantification of the flow field via PIV. To estimate a critical distance for the transition between these interaction modes, we believe that more data would be needed. Unfortunately, it is not trivial to obtain a large enough sample size of PIV data capturing live fish swimming in the laser sheet while performing our investigated behavior, and thus we think that estimating the critical distance is out of the scope of this work. From the data we have, we predict that double-sided vortex interactions require a very good in-line alignment with the foil to be able to capture vortices at both sides of the body, and we predict a critical distance within one body width from the foil center line.

12) Since fish might also swim further away (e.g., S Movie 2), what are the parameters you used to define/filter in-line swimming patterns? Line 165: You state "time their movements accordingly" to interact with the wakes generated by the airfoil -- but is it possible to add quantitative analyses of how the fish may time their movements? For example, by determining the phase difference between the airfoil motion and fish head oscillation? This seems also applicable to Line 142. Finally, in Line 146, please explain to the general reader why you did not consider using fish body wavelength for determining phase lag, which is more typical for fish kinematics. If this is not possible, please justify your processing choices accordingly, so the general reader can understand your choices and any limitations this may have for interpreting the findings.

For our analysis we considered as many swimming trials as possible that showed prolonged in-line swimming (lasting 5 to 10 seconds and more). As a result, we captured this behavior over a range of distances between fish and foil as trout naturally chose to swim at different distances. This allowed us, as suggested by the reviewers, to now include an analysis of the phase difference for different distances (see also response to comment 1). We found a linear relationship, and the parameters of the linear fit support the hypothesis of vortex interaction and synchronization (L186-198).

Thank you for mentioning the body wavelength metric. The overall phase lag along the body and the body wavelength carry similar information and can be computed one from the other (λ=2πΔΦ). Whereas phase lags are commonly reported in the field of neuroscience and robotics, wavelengths appear more often in biomechanics. To reach a larger audience we now report the wavelength in addition to the phase lag in the methods and the supplementary data file (L507-509).

13) Line 122 states you chose St around 0.267. Please outline how you selected the oscillation frequency of the airfoil with respect to the natural tail beat frequency. Also, please clarify how you selected the experiment parameters a_{sway}, a_{yaw}, amplitude, since there are different combinations of these parameters possible that give St ~ 0.267, and not all combinations are biologically representative or informative. Please include typical kinematics data of real fish here to help justify the parameter selection and, whenever there is a clear deviation, please list it as a limitation in the introduction and include it in the discussion.

We would like to refer the reviewers to our section ‘Flapping foil’ in the Materials and methods. Here we reported all the parameters including a_{sway} and a_{yaw}. We added additional explanations (L461) regarding the choice of these amplitudes (width of the wake) and the frequency (in the range of natural tail beat frequencies). Regarding the typical kinematics of real fish, we would like to point out that our goal was not to match the kinematics of the tail fin of real fish with our flapping foil. Our aim was rather to induce similar wakes with the rigid flapping foil by achieving fish-like Strouhal numbers together with the corresponding width of the wake (≈4cm). The results are illustrated in Figure S2, where we provide a direct comparison of the resulting thrust wakes.

Also, in quite a number of previous publications over the years we have published both kinematics and hydrodynamic wakes from swimming fishes and so we prefer not to duplicate that information here. The thrust wake generated by the flapping NACA 0012 very closely matches fish wakes produced by the tail. In addition, just last year we published swimming kinematics from 44 species of fishes (DiSanto et al., 2021) and provide a large data table with waveform characteristics and Strouhal numbers in addition to a number of other kinematic parameters including head and tail amplitudes for these 44 species of swimming fishes.

Di Santo, V., et al. (2021). "Convergence of undulatory swimming kinematics across a diversity of fishes." Proceedings of the National Academy of Sciences 118(49): e2113206118.

Line 35: Please clarify: "however, no data are available on live fish interacting with thrust wakes". Most of the previous studies (such as ref. 5 and 6) with real fish systems provide some data on live fish interacting with thrust wake such as kinematic data representative for the hydrodynamic interactions.

Thank you for your comment! We rephrased this sentence and modified the abstract accordingly (L36). We also added more references related to fish interacting with thrust wakes (Please see response to comment 3).

Line 129: "with the foil in a stationary position", this needs to be clarified for a general reader. Should we interpret this as the foil being in a stationary position above the water? Otherwise, should we assume it is generating a von Karman drag wake with vortices? Since these airfoils have large-scale flow separation (laminar separation bubbles) at these sub-critical Reynolds numbers (below roughly Re = 100,000 / 200,000 depending on airfoil shape). Please clarify in the methods (and discussion in case there may be interference).

We apologize for the possible confusion this term has caused. By stationary position we describe the state in which the hydrofoil is placed in the water but not moved. We have clarified this in the text now (L157). It is true that the stationary foil in the water generates a very small and narrow (due to the low drag of the NACA 0012 foil) Karman drag wake with small vortices. However, this happens on a scale several magnitudes smaller than the width of the fish tested in our experiments. In addition, we can assure the reviewer that we did not find laminar separation bubbles as the flow over the stationary foil is smooth at these Re. The main reason for having a reference condition with the foil in the water is to account for the behavioral aspect of fish interacting with any object whether it is moving or not. We wanted to have a good control condition that takes into account an object that fish could orient to regardless of its hydrodynamic properties. This allowed us to attribute the differences in fish swimming to hydrodynamic effects rather than any behavioral attraction between the fish and the foil.

Figure 2: Should Panel C not be A, since it illustrates the phase lag calculation? Please resolve.

It is correct that panel C illustrates the phase lags. However, we chose to order the panels as they appear in the text. In particular, we would like to keep panel A for the frequencies as this is the first result that we present and the order of panels is the same as in the text discussion.

Figure 3: Shouldn't the legend for the color bar be Vorticity along the z-axis?

Unfortunately, we are not entirely sure what the reviewer is referring to here. We do have a color bar on the right denoted with vorticity and the corresponding unit. The orientations are illustrated as well and are consistent with a right handed coordinate system (positive vorticity vector points out of the plane).

Line 202: "46% and 86% decrease compared to free-stream swimming" -- does this mean there are only two cases analyzed here? How did you establish these values? What is the zone around the fish head you used to evaluate pressure to make the comparison? Does this limit the analyses? Please clarify in a way the general reader can understand.

We have expanded the explanation of the computation of the head pressures in the Materials and methods section (L546-551). We now also include how we defined the zone around the fish that is used to derive the corresponding average values. It is correct that our analysis of the head pressures is at this point limited to two examples with double-sided vortex interactions and one example with single-sided vortex interactions. This is in part related to the difficulty of obtaining such samples, as fish have to swim in the laser sheet (for PIV) for multiple swimming cycles. Nonetheless, we believe that these data provide relevant insight into the underlying dynamics of fish swimming in the thrust wakes that are interesting for future modeling and experimental studies. We mention the limitations now explicitly in the discussion (L399-408).

Line 323: You claim "swimming in-line is a frequently occurring situation" -- how often do fish swim in line compared to other formations?

We appreciate the comment and have removed the word ‘frequently’ from the text in this paragraph (L411, L413) as this statement stems from our preliminary data on fish schools and needs further quantification. We have several ongoing fish schooling kinematic projects that will quantify exactly how often fish in schools position themselves in particular configurations, but at the moment we only have video data and no quantitative analyses.

References

Akhtar, I., Mittal, R., Lauder, G. V., and Drucker, E. (2007). Hydrodynamics of a biologically inspired tandem flapping foil configuration. In Theoretical and Computational Fluid Dynamics (Vol. 21, Issue 3, pp. 155–170). https://doi.org/10.1007/s00162-007-0045-2

Harvey, S. T., Muhawenimana, V., Müller, S., Wilson, C. A. M. E., and Denissenko, P. (2022). An inertial mechanism behind dynamic station holding by fish swinging in a vortex street. Scientific Reports, 12(1), 12660.

Karbasian, H. R., and Esfahani, J. A. (2017). Enhancement of propulsive performance of flapping foil by fish-like motion pattern. In Computers and Fluids (Vol. 156, pp. 305–316). https://doi.org/10.1016/j.compfluid.2017.07.016

Lauder, G. V., Anderson, E. J., Tangorra, J., and Madden, P. G. A. (2007). Fish biorobotics: kinematics and hydrodynamics of self-propulsion. The Journal of Experimental Biology, 210(Pt 16), 2767–2780.

Lauder, G. V., Lim, J., Shelton, R., Witt, C., Anderson, E., and Tangorra, J. L. (2011). Robotic Models for Studying Undulatory Locomotion in Fishes. In Marine Technology Society Journal (Vol. 45, Issue 4, pp. 41–55). https://doi.org/10.4031/mtsj.45.4.8

Li, L., Nagy, M., Graving, J. M., Bak-Coleman, J., Xie, G., and Couzin, I. D. (2020). Vortex phase matching as a strategy for schooling in robots and in fish. Nature Communications, 11(1), 5408.

Marras, S., and Porfiri, M. (2012). Fish and robots swimming together: attraction towards the robot demands biomimetic locomotion. Journal of the Royal Society, Interface / the Royal Society, 9(73), 1856–1868.

Shua, C., Liua, N., Chewa, Y., and Lub, Z. (2007). Numerical simulation of fish motion by using lattice Boltzmann-Immersed Boundary Velocity Correction Method. In Journal of Mechanical Science and Technology (Vol. 21, Issue 9, pp. 1352–1358). https://doi.org/10.1007/bf03177420

Van Buren, T., Floryan, D., Bode-Oke, A. T., Han, P., Dong, H., and Smits, A. (2019). Foil shapes for efficient fish-like propulsion. In AIAA Scitech 2019 Forum. https://doi.org/10.2514/6.2019-1379

Zhang, P., Krasner, E., Peterson, S. D., and Porfiri, M. (2019). An information-theoretic study of fish swimming in the wake of a pitching airfoil. In Physica D: Nonlinear Phenomena (Vol. 396, pp. 35–46). https://doi.org/10.1016/j.physd.2019.02.014

[Editors' note: further revisions were suggested prior to acceptance, as described below.]

The article has been improved but there are some remaining issues that need to be addressed:

The authors need to fully address the helpful feedback from the reviewers on the limitations of their hydrodynamic analysis. NACA 0012 is not an appropriate Low-Reynolds number airfoil and suffers from laminar separation bubbles at low Reynolds numbers, a low Reynolds number drag-crisis resulting in high drag, and a stall hysteresis loop. Classic low Reynolds number airfoil literature on these issues should not be ignored. That does not make the work unpublishable, but the reader needs to know about these limitations regardless of the fact that indeed many previous papers ignored them -- so that future studies can also consider more appropriate airfoils to see if that makes a difference to conclusions. Demonstrating that this has indeed an insignificant effect in the present research paradigm would require comparing a sub- and supercritical airfoil and showing that the outcome is highly similar; especially considering the lift-drag ratio of NACA 0012 is so much lower than even a flat or cambered plate at a low Reynolds number.

Previous work (Triantafyllou et al. 1993; Triantafyllou et al. 2004; Anderson et al. 1998) suggests that NACA 0012 foils are appropriate for the purpose of our study where we aim to create fish-like wakes. They demonstrate that these foils show high propulsive efficiencies if they are operated in a Strouhal number range of 0.25 to 0.35 and that the reverse Karman vortex street with its increased wake velocities is related to thrust production. In part, high efficiency is associated with leading-edge vortices that are convected downstream and interact with trailing-edge vortices that result in a reverse Karman street (Anderson et al. 1998). Furthermore, previous studies have shown that fish operate in a similar range of Strouhal numbers and also produce reverse Karman vortex streets (Saadat et al. 2017).

We acknowledge the points raised by the reviewer regarding NACA 0012 being a supercritical airfoil and that a comparison to subcritical foils (cambered, flat) could be interesting. We mention this in the Materials and methods section now. However, as mentioned above, we were able to generate thrust producing wakes characterized by reverse Karman vortex streets with increased wake velocities which were sufficient for the purpose of our experiments as they provide the flow structures that fish face in wakes of other fish.

The NACA 0012 foil, moved in the manner we programmed, generates an excellent fish-like wake. We are well aware of separation bubbles formed by NACA 0012 airfoils (see our previous paper using static testing of these airfoils which shows such bubbles (Domel et al., 2018)). Many of these studies (like this one of ours) are performed under static conditions with the airfoil stationary at each angle of attack tested. Under these conditions and at relatively low Re of 5000 to 100,000 a separation bubble will indeed form and then can peel off causing a loss of lift if it grows to a large size.

But our goals in this manuscript were to (1) use a fish-shape to (2) generate a fish-like wake. The NACA 0012 foils are excellent for this purpose as we show in this and a number of previous papers quantifying the wakes of these foils moved in heave, pitch, and both. We emphasize that in our experiments the airfoil is *moved* and that it does not matter that under these dynamic conditions that a separation bubble forms. Any separation bubble becomes incorporated into the overall wake vortex structure that is shed as the foils move back and forth.

We cite one of our previous papers with NACA 0012 foils below (Lauder et al., 2007) to show PIV data from heaving and pitching NACA 0012 foils. Movement of the airfoil in the manner that we programmed generates a fish-like wake that was the whole point of using the flapping foil.

Anderson, J. M., Streitlien, K., Barrett, D. S., and Triantafyllou, M. S. (1998). Oscillating foils of high propulsive efficiency. Journal of Fluid Mechanics, 360, 41–72.

Triantafyllou, G. S., Triantafyllou, M. S., and Grosenbaugh, M. A. (1993). Optimal Thrust Development in Oscillating Foils with Application to Fish Propulsion. Journal of Fluids and Structures, 7(2), 205–224.

Triantafyllou, M. S., Techet, A. H., and Hover, F. S. (2004). Review of experimental work in biomimetic foils. IEEE Journal of Oceanic Engineering, 29(3), 585–594.

Saadat, M., Fish, F. E., Domel, A. G., Di Santo, V., Lauder, G. V., and Haj-Hariri, H. (2017). On the rules for aquatic locomotion. In Physical Review Fluids (Vol. 2, Issue 8). https://doi.org/10.1103/physrevfluids.2.083102

Domel, A. G., Saadat, M., Weaver, J., Haj-Hariri, H., Bertoldi, K. and Lauder, G. V. (2018). Shark denticle-inspired designs for improved aerodynamics. Journal of the Royal Society Interface 15, 20170828.

Lauder, G. V., Anderson, E. J., Tangorra, J. and Madden, P. G. A. (2007). Fish biorobotics: kinematics and hydrodynamics of self-propulsion. Journal of Experimental Biology 210, 2767-2780.

Also, while wall effects may be acceptable for answering the present research question, given how close the walls are, there needs to be a more detailed discussion of possible limitations. Wall interference is characterized based on airfoil/ propulsive body length, not width. For largely separated flow phenomena, such as shed vortices, there can be a marked wall effect when they are this close. It may not invalidate the work, but the cited literature ("In addition, wall effects (Webb, 1993) have minor to no measurable effects on fish swimming in our experiments.") lacks the fluid mechanic depth to argue that there is no measurable effect. That said, the experiment does seem representative of fish swimming close to a solid boundary, and for those conditions, the findings appear robust. There is no reason to believe that the non-quantified wall effect would disqualify the work for understanding how fish swim further away from a wall. But eLife offers the space to report that wall effects were not quantified (in the way a fluid mechanist would expect this to be done; boundary layer thickness at the wall is not actually sufficient for this). Given how close the walls are in body lengths this requires more consideration in future work, for example, with a robot fish mounted on a force sensor. The authors do not need to do this work for inclusion in the present article, if claims are tempered appropriately.

Wall effects is actually a topic that we have previously investigated in some detail using the approach recommended above: using an experimental system with force measurement to quantify locomotor forces during propulsion at varying distances from the walls of the flow tank, and with both kinematic and PIV analyses of flow patterns near the wall. One relevant paper is Quinn et al. (2014a, citation below) and this study was actually performed in the *same* flow tank and with the *same* robotic flapper used for this manuscript. We used a 6-axis force/torque sensor to quantify propulsion speed, forces, and efficiency at varying distances from the wall.

The results of this analysis, discussed below in some detail, show why we are very confident that we are not experiencing significant wall effects in our current manuscript.

In brief, here are the results from that previous study Quinn et al. (2014a) paper to illustrate our key response points. Quinn et al. studied the propulsion of relatively large flexible panels (150mm span by 195 mm length) at varying distances from both the wall and bottom of the flow tank. Please note that this panel size is much larger in area than the trout studied here and that the panels, when moved, came as close as 15mm from the wall which is much, much closer than the trout studied here which were in the middle of the tank (approx. 30 cm on a side and 1 meter long working area for filming). Plastic panels of three flexural stiffnesses were studied (see Table 1 Quinn et al. (2014a)) and these panels were moved at the leading edge (in a manner similar to the foil motion used in this manuscript) to generate a propulsive wave. The flexural stiffness of the panels in Quinn et al. (2014) was specifically chosen to encompass the range of actual fish flexural stiffness which range from 10–3 to 10-6 (Shelton et al., 2014) and Panel B (Table 1 Quinn et al. (2014a) ) matches nicely the flexural stiffness of trout bodies.

Now, Quinn et al., 2014a: Figure 5 shows the effect on propulsive speed and economy at different distances from the wall, given as the parameter d/a where d=mean distance from the wall, and a=heave amplitude of the flexible body at the leading edge. Note that there is minimal to no effect of distance on economy and little to no effect of distance on self-propelled speed for all panels at d/a >5. Please note also that for the most trout-like panel (B) there is almost no effect at any distance.

In this manuscript, trout operated at a value of between 7 and 3.5 depending on whether “a” is taken as head oscillation or tail oscillation amplitude.

Although swimming close to a surface with an undulatory body can certainly improve propulsive efficiency and alter the time-dependent profiles of forces and torques, we believe that these previous experiments using the exact same experimental system show that it is highly unlikely that wall effects have influenced our results for trout swimming in the center of the flow tank.

One final study from Quinn et al. (2014b) used another flapper/flow tank system at Princeton Univ. to investigate the wall effect for a rigid pitching foil at different distances from the wall. Their data show that for conditions similar to that of our trout experiments where trout are 15 cm from the wall, that there is no effect on propulsive forces.

Overall, we would like to emphasize as these studies show, that the motion of flexible bodies near surfaces (particularly under conditions of self-propulsion where free-stream flow convects vortices downstream away from the undulatory surface) has some unexpected results which are not well represented by classical static airfoils studied near a wall.

One last point. A second study from our lab (Blevins and Lauder, 2013) also analyzed ground/wall effects using a flexible propulsor (again in the *same* flow tank and with the *same* robotic flapper system). That study quantified self-propelled speed, flow patterns near the wall with PIV, and kinematics, and calculated the cost of transport (using data from a force transducer) comparing undulatory propulsion in the center of the flow tank to a position where the undulatory body approached within 1 cm of the wall. This study found some small differences in the cost of transport at some actuation frequencies, but that at least two actuation patterns show no difference.

Again, the trout in our experiments swam in the center of the flow tank, and not near any wall. So from these results also we expect that the propulsion of our trout was not in any way altered by a “wall effect”.

Quinn, D. B., Lauder, G. V. and Smits, A. J. (2014a). Flexible propulsors in ground effect. Bioinspiration and Biomimetics 9, 1-9.

Quinn, D. B., Moored, K. W., Dewey, P. A. and Smits, A. J. (2014b). Unsteady propulsion near a solid boundary. Journal of Fluid Mechanics 742, 152-170.

Blevins, E. L. and Lauder, G. V. (2013). Swimming near the substrate: a simple robotic model of stingray locomotion. Bioinspiration and Biomimetics 8, 016005.

Shelton, R. M., Thornycroft, P. J. M. and Lauder, G. V. (2014). Undulatory locomotion by flexible foils as biomimetic models for understanding fish propulsion. Journal of Experimental Biology 217, 2110-2120.

https://doi.org/10.7554/eLife.81392.sa2

Article and author information

Author details

  1. Robin Thandiackal

    Harvard University, Cambridge, United States
    Contribution
    Conceptualization, Data curation, Software, Formal analysis, Investigation, Visualization, Methodology, Writing - original draft, Project administration, Writing - review and editing
    For correspondence
    rthandiackal@fas.harvard.edu
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-8201-4892

  2. George Lauder

    Harvard University, Cambridge, United States
    Contribution
    Conceptualization, Supervision, Funding acquisition, Methodology, Writing - review and editing
    For correspondence
    glauder@oeb.harvard.edu
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-0731-286X

Funding

National Science Foundation (EFRI-830881)

  • George Lauder

Office of Naval Research (N00014-18-1-2673)

  • George Lauder

Office of Naval Research (N00014-14-1-0533)

  • George Lauder

Office of Naval Research (N00014-21-1-2210)

  • George Lauder

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

This work was supported by the National Science Foundation (Grant number EFRI-830881) and the Office of Naval Research (Grants N00014-18-1-2673, N00014-14-1-0533, and N00014-21-1-2210). We thank members of the Lauder Lab for many helpful discussions about in-line swimming, and Prof. Valentina DiSanto for collaborative research on Menidia schooling shown in Video 6. Publication charges paid by a grant from the Wetmore Colles fund, Museum of Comparative Zoology, Harvard University.

Ethics

All experiments were performed in accordance with Harvard animal care and use guidelines, IACUC protocol number 20-03-3 to George V. Lauder.

Senior Editor

  1. Christian Rutz, University of St Andrews, United Kingdom

Reviewing Editor

  1. David Lentink, University of Groningen, Netherlands

Reviewer

  1. Maurizio Porfiri, New York University, United States

Version history

  1. Received: June 25, 2022
  2. Preprint posted: July 15, 2022 (view preprint)
  3. Accepted: February 3, 2023
  4. Accepted Manuscript published: February 6, 2023 (version 1)
  5. Version of Record published: March 22, 2023 (version 2)

Copyright

© 2023, Thandiackal and Lauder

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

Metrics

  • 1,043
    Page views
  • 208
    Downloads
  • 8
    Citations

Article citation count generated by polling the highest count across the following sources: Crossref, PubMed Central, Scopus.

Download links

A two-part list of links to download the article, or parts of the article, in various formats.

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

  1. Robin Thandiackal
  2. George Lauder
(2023)
In-line swimming dynamics revealed by fish interacting with a robotic mechanism
eLife 12:e81392.
https://doi.org/10.7554/eLife.81392

Categories and tags

Further reading

    1. Physics of Living Systems

    Animal Locomotion: The benefits of swimming together

    Iain D Couzin, Liang Li
    Insight

    When a fish beats its tail, it produces vortices in the water that other fish could take advantage of to save energy while swimming.

    1. Computational and Systems Biology
    2. Physics of Living Systems

    Beta-cell intrinsic dynamics rather than gap junction structure dictates subpopulations in the islet functional network

    Jennifer K Briggs, Anne Gresch ... Richard KP Benninger
    Research Article

    Diabetes is caused by the inability of electrically coupled, functionally heterogeneous -cells within the pancreatic islet to provide adequate insulin secretion. Functional networks have been used to represent synchronized oscillatory [Ca2+] dynamics and to study -cell subpopulations, which play an important role in driving islet function. The mechanism by which highly synchronized -cell subpopulations drive islet function is unclear. We used experimental and computational techniques to investigate the relationship between functional networks, structural (gap-junction) networks, and intrinsic -cell dynamics in slow and fast oscillating islets. Highly synchronized subpopulations in the functional network were differentiated by intrinsic dynamics, including metabolic activity and KATP channel conductance, more than structural coupling. Consistent with this, intrinsic dynamics were more predictive of high synchronization in the islet functional network as compared to high levels of structural coupling. Finally, dysfunction of gap junctions, which can occur in diabetes, caused decreases in the efficiency and clustering of the functional network. These results indicate that intrinsic dynamics rather than structure drive connections in the functional network and highly synchronized subpopulations, but gap junctions are still essential for overall network efficiency. These findings deepen our interpretation of functional networks and the formation of functional sub-populations in dynamic tissues such as the islet.

    1. Physics of Living Systems

    The Reissner fiber under tension in vivo shows dynamic interaction with ciliated cells contacting the cerebrospinal fluid

    Celine Bellegarda, Guillaume Zavard ... Claire Wyart
    Research Article Updated

    The Reissner fiber (RF) is an acellular thread positioned in the midline of the central canal that aggregates thanks to the beating of numerous cilia from ependymal radial glial cells (ERGs) generating flow in the central canal of the spinal cord. RF together with cerebrospinal fluid (CSF)-contacting neurons (CSF-cNs) form an axial sensory system detecting curvature. How RF, CSF-cNs and the multitude of motile cilia from ERGs interact in vivo appears critical for maintenance of RF and sensory functions of CSF-cNs to keep a straight body axis, but is not well-understood. Using in vivo imaging in larval zebrafish, we show that RF is under tension and resonates dorsoventrally. Focal RF ablations trigger retraction and relaxation of the fiber’s cut ends, with larger retraction speeds for rostral ablations. We built a mechanical model that estimates RF stress diffusion coefficient D at 5 mm2/s and reveals that tension builds up rostrally along the fiber. After RF ablation, spontaneous CSF-cN activity decreased and ciliary motility changed, suggesting physical interactions between RF and cilia projecting into the central canal. We observed that motile cilia were caudally-tilted and frequently interacted with RF. We propose that the numerous ependymal motile monocilia contribute to RF’s heterogenous tension via weak interactions. Our work demonstrates that under tension, the Reissner fiber dynamically interacts with motile cilia generating CSF flow and spinal sensory neurons.