Effects of boundaries for high Reynolds number artificial swimmers

Contents

One fish Increasing the number of fish robots in the arena Swirling state Numerical simulations Flexible mobile arenas

We start by focusing on a single swimming fish robot both in a large tank and in circular arenas. Then we characterize the effect of robot number on the organization of the assembly. We noted that the spatial organization of the assembly of fish robots is very sensitive to the type of boundary. We have used two different arenas, a glass cylinder and a porous one made of a net like boundary with pores of a few millimeters. The arenas are deposited in a larger tank with a water layer of a few centimeters in depth (5 cm) slightly higher than the height (3.4 cm) of the fish robot so that the fish swim without touching the bottom of the tank. The rationale behind the choice of a closed and porous boundaries is to act on the flow generated by the fish inside the arena in which they are enclosed. These entities, when swimming, create a large back flow in the enclosure. While the solid boundaries can dissipate this flow, it persists for long times nonetheless. The porous boundaries on the other hand allow this flow generated by the fish to escape the arena through the pores and into the surrounding water tank thus minimizing its effect. The other reason is that the fish can bounce somewhat on the walls of the arena. But we noted that this bouncing is more important for the solid glass boundary than for the net like softer boundary. Both effects can have important consequences on the spatial organization of the fish robots under confinement, effects we will study below.

One fish

Our study focuses on fish like objects, Fig. 1 a and b, propelled by a caudal fin that oscillates at a frequency of several Hertz, see Fig. 1 c. The velocity of such fish can reach values of up to 15 cm/s, Fig. 1 d, giving Reynolds numbers (\(Re=UL/\nu\) where U is the mean velocity of the fish, L its length and \(\nu\) is the kinematic viscosity of the fluid used which is water) of approximately \(10^4\). The making of these toy fish robots, which are commercially available, gives them the ability to change the beating frequency of their caudal fin in what seems to be a random pattern. This pattern, which lasts a few minutes (4 minutes roughly), then repeats several times while the fish is immersed in water. This gives the fish what looks like a random cruise velocity, at least for periods of time less than the repeat period of the frequency pattern.

The first characterization we carried out is the determination of the flapping frequency and amplitude of the caudal fin of the fish. From video recordings of the caudal fin when the fish is held fixed but immersed in water, we find that this frequency comes in three values: 7.7 Hz, 4 Hz, and 2 Hz. While the large and medium frequencies correspond to periodic back and forth beating with the fin going from left to right and back symmetrically, the smallest one corresponds to asymmetrical beating with the fin going from left to right to left followed by a quiet period or from right to left to right followed by another rest period. These two modes at low frequency allow the fish robot to turn right or left respectively. A plot of the beating amplitude of the end of the caudal fin versus time is shown in Fig. 1 c. The amplitudes of the three different frequencies are not exactly the same but they are all of roughly 1 centimeter. The different frequencies are clearly visible along with the half duty cycle ones.

A single fish is then followed using particle tracking over a certain period of time, generally over 5 minutes in circular arenas but only 1 min or so far from any boundary in a large tank, to determine the fish position, its orientation, and its velocity. The fish have been painted with a black rectangle on their back, as can be seen in Fig. 1 a and b, so that their orientation and position can be obtained easily from black and white images of the arena. From such determinations, the instantaneous position of the fish can be followed over time. That the three frequencies are present is seen in the velocity time trace of a fish robot swimming in a large tank far from any boundary showing a fast velocity at near 17 cm/s, an intermediate velocity near 12 cm/s, and a smaller velocity near 8 cm/s, Fig. 1 d.

Next, we put a single fish in a circular arena and track its position versus time, Fig.2 a and b. This information is then used to extract the velocity time trace and velocity distribution as well as the probability of presence versus the radial distance from the center of the arena. Not surprisingly, the fish robot spends most of its time swimming along the boundary. It does escape from time to time but ends up returning to the wall area. The trajectory of the fish robot can be seen in Fig. 2a and b where most of the time, the fish robot swims along the arena wall.

The distribution of radial position obtained from this trajectory is thus peaked near the wall at roughly a fish robot width, and this for both arenas: for a glass cylinder and a porous plastic cylinder. From the histogram of velocity (total velocity), the distribution is wide but three different peaks at roughly 10–12 cm/s, 8-7 cm/s, and 5-4 cm/s seem to be apparent corresponding to the three different frequencies of beating of the caudal fin. These velocities are smaller than the ones of Fig. 1 where the fish robot was far from boundaries. The proximity of the boundary reduces the speed somewhat.

Overall, the fish robot swims near the boundaries over the course of their swimming. This feature is reminiscent of what occurs for low Reynolds number swimmers²⁶or dry rod like particles^27,28,29.

An interesting feature which distinguishes the two different boundaries is the angle of the fish robot with respect to the wall. As can be seen in the images of Fig. 2 c and d, the fish robot makes a smaller angle with the wall for the impermeable boundary compared to the porous boundary.

Figure 2 e shows the distribution of angles for the two arenas. While the distribution shows a peak centered at an angle of roughly \(10^{\circ }\) or less for the impermeable case, the peak of the distribution is centered at a larger angle for the porous case. This angle is roughly \(22^{\circ }\) in this case. As we will see below, this feature has direct consequences on the spatial organization of the fish in the two different arenas as we will see below.

Increasing the number of fish robots in the arena

Let us now consider the role of the number of fish robots in the arena. As the number of robots increases, their spatial organization evolves. For a few fish robots, the robots continue swimming along the boundary and the radial distribution is peaked near the wall. While some features (radial distribution and velocity distribution) seem to be barely dependent on the boundary (solid or porous), as they remain roughly the same, an intriguing development occurs as can be seen in Fig. 3a and b for as little as 3 fish robots. Contrary to the impermeable boundary where no particular organization emerges apart from the fish robots swimming near the boundary and being more parallel to it (Fig. 3c), for the porous walls, the three fish robots self-organize rather quickly. They adopt a configuration where they all go around the arena in the same direction while maintaining a configuration where the three are parallel to each other and are at a slightly slanted angle with respect to the wall. This state of organization can persist for several minutes. If it breaks, the fish robots have a tendency to come back to this configuration rather quickly within tens of seconds. This is a direct consequence, we believe, of the angle at the wall of a single fish robot as we have observed above in Fig.2 e. This angle is roughly \(22^{\circ }\) as can be seen in Fig. 3 c where the distribution of angles for the 3 robots is shown under different conditions.

For the glass arena, the three fish robots do not necessarily end up in the configuration described above. This configuration can occur, but does not last for long. Once it breaks, the fish robots take a long time before recovering this configuration with all fish robots swimming in the same direction. Generally, for the glass arena, the fish robots seem to make a small angle with respect to the wall and follow each other head to tail as seen in Fig. 3 a. The difference with the porous boundary stems from how the robots algn with the wall as seen in Fig.2 for 1 robot and Fig.3 c for the assembly of 3 robots where the most probable angle is close to zero for the impermeable cases.

An interesting and crucial observation is that the angle that the fish robots make with the boundary has a different distribution when the boundary changes. The fish robots are more parallel to the boundary for the closed one (glass impermeable boundary) but have a more tilted angle for the porous one. As the angle is more tilted, a configuration of 3 or more robots that are parallel to each other forms a small stable cluster as observed for the 3 fish robots above in Fig. 3 b. For the impermeable boundary, the fish robots make a small angle with respect to the the wall but they do not swim necessarily parallel to each other: they form a line of fish robots in a head to tail configuration where the follow each other, Fig. 3 a. Because of small collisions with the boundary and the fact that the fish robots can have different speeds, the robot ahead of the two others can bounce on the boundary and change course. As this happens, the second fish robot right behind it can overtake it and move upfront followed by the third robot. The two continue their way forward while the first one is pushed out of the boundary region and may change direction. The time for it to come back can be long. This difference, even at low fish robot number, has consequences on what occurs for a larger number of robots in the arena as we will see below. A test using a porous boundary surrounded by a glass cylinder seems to give results close to those obtained with the glass boundary (see Fig. 3 c) highlighting the effect of porosity instead of simply the presence of the pores and the associated roughness of the surface.

To distinguish among the different organizations and to further study the differences between porous and non porous walls, we calculate the normalized angular momentum of the assembly. This can be written as \(\tilde{L}= \frac{\sum {\textbf {r(t)}}\textbf{x} {\textbf {V(t)}}}{VR}\). Here \({\textbf {V(t)}}\) is the velocity vector of the fish robot, \({\textbf {r(t)}}\) its radial position vector (from the center of the arena), V is the mean velocity of a single fish robot in a circular arena (which is 9 cm/s) and R the radius of the arena. \(\tilde{L}\) is a vector but it only has one component in the vertical (or vorticity) direction since the layer is quasi two dimensional. The sum is carried out over all agents in the arena. This quantity is stochastic (see Fig. 3 d) as it is calculated for each image and its distribution is calculated for each configuration (porous or impermeable). If all fish robots are near and parallel to the boundary, and swim in the same direction, the value of \(\tilde{L}\) is maximum with a value close to 1 (or −1). If the fish robots swim in different directions, or their positions are far from the wall, this angular momentum is smaller.

A time series of the angular momentum for 3 fish robots is shown in Fig. 3 d. Note that for the porous boundary, \(\tilde{L}\) fluctuates around a nonzero value while for the impermeable boundary the value of \(\tilde{L}\) can change sign and go through zero from time to time. The angular momentum (the absolute value of \(\tilde{L}\)) distribution for the 3 fish robots in the porous arena, Fig. 3 e, is peaked at a non zero value. Visual inspection shows that the 3 robots turn in the same direction and remain close to the wall which is consistent with the non-zero value of the angular momentum. We will refer to this behavior as swirling. Generally, the 3 fish robots would form a stable cluster with a finite angle with respect to the wall and swim in the same direction for a relatively long period of time as illustrated by the example of Fig. 3 b. However when the boundary changes to impermeable, the distribution of the angular momentum is no longer peaked at a finite non zero value. The distribution of the absolute value of L, Fig. 3 e, has a broad shape and an important component at zero angular momentum. In this case, swirling is less important with events where the fish robots no longer have the same direction becoming more and more important. This is the first sign that the boundary plays a role in how the fish robots self organize in the arena even for small numbers.

Swirling state

The distributions of velocity and angular momentum discussed above for a few fish robots are now examined for the two types of boundary for more robots. Figure 4 shows how these distributions vary with the number of fish in the same arena.

Consider the velocity distributions, Fig. 4a and b. In both arenas the velocity distribution is peaked at a velocity which depends on the number of fish N. While for small numbers (less or equal to 3) the shape and the peak position of the distribution do not change (for glass), the peak position starts to shift gradually to smaller and smaller values as N increases. For the porous walls a roughly similar phenomenology is observed. As the number of fish robots increases, a gradual slowing down of the robot assembly takes place independently of the nature of the boundaries.

Concerning the angular momentum, Fig. 4 c and d, as the number of fish robots increases, the distribution goes from a distribution with a peak far from zero to a distribution peaked around zero. The mean value varies thus from high values at low densities to close to zero at higher densities. This variation is more severe for the impermeable boundary where the distributions have a peak near zero as soon as the number of fish robots increases above 2 or 3. The distributions of \(\tilde{L}\) for the porous boundary continue to have a peak away from zero for values of N as high as 18. Note that these changes occur while the velocity distributions remain roughly similar for the two arenas.

In Fig. 4e and f, we plot the mean value of the velocity and mean value of the angular momentum. The mean velocity decreases with the number of fish robots roughly similarly for the two types of boundary; the mean velocity is slightly smaller for the glass walls probably because friction is more important compared to the porous boundary where there is a certain amount of slip probably. Note that this decrease of the mean velocity with the number of robots is the hallmark of the mobility induced phase separation characteristic of active matter³⁰ which we report here for for high Reynolds number swimmers.

The mean angular momentum variation versus N depends however more strongly on the type of boundary, Fig. 4 f. It decreases rather abruptly for the glass boundary at N close to 3 while the decrease is more gradual for the porous boundary leaving a wide range of number densities for which the angular momentum is larger for the porous boundary.

We will call the state with a non zero peak position for the angular momentum distribution, a swirling state. In this state most fish robots go around the arena in the same direction and swim close to each other. In Fig. 4 g, the proportion of fish robots swimming in the same direction remains high for the porous boundary while for the solid/impermeable boundary this proportion drops to zero very quickly (\(N>3\)). This is quantified by calculating \(I=|(n^+-n^-)|/(n^++n^-)\) where \(n^+\) or \(n^-\) are the proportion of fish robots swimming in the clockwise or counter-clockwise direction. This quantity falls abruptly for the glass arena while it drops slowly and remains high up to almost 20 fish robots for the porous boundary.

The most probable configuration of the fish robots for the high angular momentum state or swirling state seems to be the parallel configuration where the fish form a linear cluster with robots parallel to each other with a slight angle with the boundary as seen for the three fish robots above (Fig.3 b). For the glass boundary, the state of organization is more disordered with fish robots swimming in different directions giving rise to a smaller angular momentum.

While the swirling state with fish robots parallel to each other can persist for the porous boundary up to several fish, it does not for the solid boundary (apart from the trivial case of 1 and eventually 2 robots). The stability of the most probable configuration is thus highly dependent on the properties of the boundary with a more disordered state for the closed (glass) boundary. Note that this difference is related to the mean angle the fish robots make with the boundary. Figure 5 shows a montage of the fish robots configuration with the solid/impermeable (Fig. 5a) and porous (Fig. 5b) boundaries: Note that for the impermeable boundary the fish robots are more randomly placed and even if the group seems to travel roughly in the same direction (see arrow), the collective movement is very slow. For the porous boundary, on the other hand, the fish form a linear cluster where the fish are rather parallel to each other and are all at a slight angle with respect to the wall. This angle is roughly of \(20^{\circ }\) as seen in the pdfs of the angle with the wall plotted earlier. The linear cluster allows the fish robots to undergo a global rotation as they all turn in the same direction (see arrow) with a full revolution in less than 20 s. Fig. 5 c (left photo) shows a snapshot of 20 fish robots most of which are in this configuration. For more robots and this is so for both arenas, the fish robots form disorganized clusters that barely move in the arena as can be seen in the snap shot of Fig. 5c (right photo) and time montage of Fig. 5 d for 38 fish robots in a porous arena of 30 cm diameter.

Numerical simulations

To corroborate these experimental results, we have carried out direct numerical simulations (DNS) of the Navier Stokes equations of objects whose shape is similar or close to the fish robots used in the experiments^25,31 and immersed in a shallow fluid layer, Fig.6a, b, and c. These simulations use the penalization method³² and both porous and impermeable walls are used. Further the shallow layer of fluid used is delimited by a solid bottom and a frictional top to account for the presence of the tank bottom and the free surface of the fluid. Also, these objects are propelled by the oscillations of the caudal fin which is programmed to mimic the experimental oscillations with roughly similar amplitudes and frequencies.

The incompressible Navier-Stokes equations, which describe the dynamics of viscous, incompressible fluids, are discretized in time using a fractional time-step method of the prediction-correction type, as proposed by Chorin and Temam^33,34. This method is commonly used to solve the Navier-Stokes equations in a stable and efficient manner by splitting the velocity update into two steps: a first prediction step that computes a velocity field from a given pressure field, followed by a correction step to adjust this prediction and ensure the incompressibility and viscosity conditions are satisfied.

The spatial discretization is carried out using the finite difference method on a uniform Cartesian grid. This choice ensures ease of implementation while efficiently handling complex fluid problems. Convective terms, which represent the transport effects within the fluid, are discretized using a third-order upwind scheme, which is particularly well-suited for advection equations to avoid non-physical oscillations (such as ”overshooting”) and ensure better stability. Other terms, such as diffusion or pressure-related terms, are handled using second-order finite differences, providing a good balance between accuracy and computational cost. Some details can be found in references^25,31,35.

In this simulation framework, immersed structures (such as swimmers) are modeled within the fluid. This requires a specific approach to handle the interaction between the fluid and these structures. The immersed boundary method³⁶is used to represent these structures without explicitly modifying the mesh around them. This approach simplifies the computation while maintaining a good approximation of the boundary conditions imposed by the solid structures. This approach is coupled to the Volume Penalization approach to simplify the imposition of pressure boundary conditions on the solid boundaries³⁷.

The swimmer’s surface is discretized using specific meshes, where forces and moments due to the fluid-structure interaction are calculated. These forces are then integrated over the swimmer surface to evaluate the swimmer’s dynamic behavior within the fluid. This includes the consideration of lift forces, drag forces, and other hydrodynamic forces that influence the swimmer’s motion. The method also accounts for changes in the shape of the swimmer in response to forces exerted by the fluid, allowing for simulations of complex interactions between the swimmer and the surrounding fluid³⁵.

A key feature of this approach is the introduction of contact forces to handle collisions, both between swimmers themselves and between swimmers and surrounding walls. Managing these collisions is essential for simulating realistic interactions in environments where multiple solid structures may move independently and come into contact with each other. Lubrication and collision forces, developed in^38,39, are employed to model the short-range interactions between swimmers, as well as between swimmers and walls.

The numerical solver has been validated through a series of varied test cases and practical applications, ensuring its robustness and accuracy. These tests include simulations of fluid-structure interactions as well as comparisons with analytical or experimental results when available^35,40. The results demonstrate that the proposed method is capable of accurately capturing the effects of fluid-structure interaction, particularly in both low and high-speed regimes, and is able to handle complex geometries, such as those encountered in swimmer modeling presented in this study.

Figures 6 d and e show the results of the trajectory, velocity distribution, and probability of presence for a single numerical fish. Note that the fish remains near the wall for most of its trajectory as observed experimentally. The velocity time trace shows the different velocity regimes and the changes associated with the changing frequency of the beating of the caudal fin. The pdf of the velocity shows similar features as the experiments with a broad distribution and the presence of characteristic velocities associated with the three characteristic beating frequencies of the caudal fin. No major differences are observed for a single numerical fish between the porous and impermeable boundaries. A snapshot of these simulations is shown in Fig. 6b and c where the fish is near the boundary and the velocity field in the mid-plane of the shallow layer is shown. Note that for the porous boundary the flow field is present outside the arena as well. Note that a major difference between the two arenas is the wake behind the fish: one of the vortex streets behind the caudal fin is absent near the boundary for the impermeable one while both are present for the porous boundary. Note that these streets emerge when the caudal fin beats. The beating from right to left produces clockwise rotating vortices while the beating from left to right produces counterclockwise rotating vortices. This produces a reverse von Karman street¹². Along these features we have determined the angle at the wall of the numerical fish and in this case a difference appears depending on the nature of the wall as we have observed in experiments. Figure 6 f shows the distribution of the angle the fish make with the wall. For the porous boundary, the distribution is peaked at an angle of roughly 0.22 radians (12/13 deg). This angle is compatible with the angle observed in experiments albeit smaller. For the impermeable boundary the distribution has a peak relatively close to zero (less than \(6^{\circ }\)) and many events very near zero as the fish is basically parallel to the wall. Both features are reminiscent of the experimental findings.

We believe that this difference in angle with respect to the wall is a consequence of the differences in the wake structure of the fish near the boundaries. As noted above, the caudal fin produces vortices when beating. These are produced near the tip of the caudal fin, and their direction of rotation depends on the beating direction. In fact the vortex street produced is referred to as a reverse von Karman street since the vortices behind the fin rotate in the opposite direction compared to a standard vortex street behind a static obstacle. These vortices can produce a drag on the tip as do wing tip vortices. This drag will alternate direction depending on the direction of rotation of the vortices. For the caudal fin stroke, the vortex will produce a force in the same direction as the direction of the stroke. Near the wall the fin beating towards the wall will experience a force pointing towards the wall. The absence of the vortices in the stroke away from the wall as observed for the impermeable boundary leaves a force imbalance where the tail is only pushed towards the wall giving rise to a small angle. For the porous boundary each stroke both towards the wall and away from the wall produces a force due vortices giving no net force on the tail. This difference in the flow details between the two boundaries gives rise to the differences in angle and as we have seen above to large differences in how the fish organize near the walls. The numerics give a fine and quantitative picture of the flow details and thus allow a physical understanding of the role of the boundaries and their nature on the behavior of the robo-fish.

In a series of tests where 3 and 9 fish are used, different observations can be made. First of all, the distribution of the positions of the fish is basically constrained in a region very close to the wall as seen in Fig. 7 for nine fish. Second, the velocity distribution is broad but its peak shifts to smaller values as the number of fish increases as seen in comparing Figure 6 (e and d) and 7 (a and b) where the results for 1 and 9 fish are shown respectively. Apart from the differences in angle at the wall, it is difficult to distinguish between the two boundaries in the numerics as it is difficult to obtain enough statistics for the 9 fish case to see a difference in the angular momentum for example. For larger numbers of fish, the time duration becomes even smaller. The computational costs of this numerical simulation are considerable due to the extensive hardware requirements and large volume of data generated. A typical simulation runs for five days on a cluster of 256 CPUs, using a computational mesh with of 500x500x100 grid cells. While memory is not the primary limiting factor for the calculations themselves, the storage required to save the results is substantial. In particular, storing the output data – necessary for creating videos of the simulation – requires around 200 To per run, and sometimes even more to save the velocity and pressure fields as well as the fish positions allowing to generate the movies seen in the SI.

Nevertheless and considering the main features of the organization of the fish in the arena such as their probability of presence which is peaked near the walls and distribution of angles at the wall, as well as their velocity distributions, the numerics are in excellent agreement with the experiments. Further and even if the statistics are not sufficient, the velocity distribution shifts to smaller values as the number of fish increases again in very good agreement with experiments.

Flexible mobile arenas

While in the results presented so far the arena was fixed and rigid, in an extension of this work we have used free to move and flexible arenas as shown in Fig. 8. In the first series of snapshots (Fig. 8 a) we show that the ensemble of fish robots may self-organize to exert a common push on the arena walls and push it forward over a distance of roughly two diameters in a matter of tens of seconds. In the montage Fig. 8 b, a much more flexible arena is used. Here again the fish robots self organize to exert a collective pushing on the walls both moving and deforming the arena. When the organization of the fish is perturbed the pushing ceases and new deformations start elsewhere.

Similar properties have been uncovered in other active matter systems both numerically and experimentally at different spatial scales giving the premises of flexible robotics based on collective dynamics at different scales both in aqueous or wet environments as well as dry ones^27,28,41,42 We have simulated this behavior albeit for a 2D system as can be seen in the montages of Fig. 8c and d. The first montage shows a moving permeable arena and as it is permeable one can see the fluid flow generated by the fish leaking out of the arena. The second montage shows a flexible arena with the fish pushing on all sides and engendering large deformations. The simulations seem to capture the possibility that the fish may deform and move an arena or scaffold in agreement with the experiments.

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