Force Analysis on Swimming Bodies at Stable Configurations

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1 Force Analysis on Swimming Bodies at Stable Configurations Nickolas Lewis Courant Institute of Mathematical Sciences New York University Abstract It has been observed in nature that migratory birds fly in V-formations and each bird produces flows in the fluid which tailing birds are able to interact with. A bird following behind another in V-formation will try and trace its wing tip in the path designated by the bird in front of it in order to reduce the total amount of energy that it must expend on its journey [1]. While this is the prevailing theory, this study aims to formulate a new model on flight formation based off of the work by Sir James Lighthill, who hypothesized that for sufficiently fast locomotion,these formations could arise naturally due only to aerodynamic interactions, without the need for communication between birds []. Such interactions could lead to a lattice structure of formation in the form of a V with birds locked into stable configurations in this formation, akin to atoms in a lattice. We test the validity of this model of birds and flight formation with wing foils submerged in a cylindrical tank. 1 Introduction Over the past couple decades, the mechanisms governing V-Formation flight of migratory birds have been studied in detail. Peter Lissaman and Carl Schollenberger were the first to theoretically predict the optimal shape of avian flight formation [3], while others aimed to devise models of the fluid interactions happening in the wake of each bird. Over forty years later, a study by Steven J. Portugal provided a degree of experimental verification to Lissaman and Schollenberger s results by applying circular statistical analyses to the formation flight of Northern Bald Ibises. Portugal observed that when in the wake of a leader in V-Formation flight, a follower bird would attempt to trace out the same path in space as the leading bird as to benefit off of the upwash produced by the free trailing tip vortices [1, 4]. While this is the prevailing theory, this study attempts to test the validity of the other hypothesis, the Lighthill Conjecture, and model the forces governing the flow interactions described by it. Sir James Lighthill hypothesized that for sufficiently fast locomotion, aerodynamic interactions between birds could serve as a means of naturally configuring them into formations []. This structure is analogous to the lattice structures found in crystals. By better understanding the fluid interactions 1

2 among swimmers and how they determine these stable configurations, the results may serve as an approximation for birds in actual formation flight. In this research, we aim to study the two-body and three-body problems. By studying the threebody problem, we aim to show that the fluid interactions produced by a leader on a follower can be modeled as a pairwise interaction between only those two swimmers. By doing so, we eventually hope to apply these results to arbitrarily many bodies. Experimental Setup To measure flow interactions between individual pairs of swimmers, a system was designed consisting of cylindrical bearings attached to a vertical axle with bars attached to each bearing with a wing foil of length l = 8cm at the end of each bar, as shown in the Figure 1. Wings were designed to have a trailing edge, defined by the sharpness of the edge, and a leading edge, defined by the roundness of the edge. Such a system allows for up to five wing foils, however, only configurations of two and three foils were tested in this experiment. The two wing configuration was achieved by simply removing the rod on the bottom bearing. Each wing foil is attached to a steel rod which is attached to low friction rotary bearings, with the friction coefficient being approximately the same for each bearing, and bent so that each wing would be on the same horizontal plane as each other. Each wing was then bent clockwise or counterclockwise in order to eliminate any angle of attack α. In other words, the angle of attack of each wing W i is α i. Each bar was radially.5cm out from the center of the tank. Axle Wing Foils Bearings 3 1 Figure 1: A motor thrusts the wings up and down by an amplitude A with a prescribed frequency f in a cylindrical tank. This generates thrust which allows for the wings to swim. The wing foils were placed in a cylindrical tank and heaved up and down a prescribed vertical displacement by a motor, or in other words an amplitude. The

3 prescribed amplitudes in this experiment were A 1 = 1.5cm and A =.cm. Another motor provided this heaving motion with a frequency of f =.5Hz. The heaving motion of the axle provided thrust for each wing due to their shape, allowing it to rotate counterclockwise in the tank. The distance between two swimmers was defined as G, the gap distance, which was the length of the arc between the center of one swimmer s trailing edge to the center of the leading edge of the follower directly behind it. Stable positions were denoted by G i for the i-th stable position where i is a positive integer. These positions were determined by allowing the wing foils to swim for a period of 3 s. Wings were constantly perturbed in order to examine whether or not they would return to their original position in the configuration. A high speed camera along with image processing software in MATLAB were used to determine the gap distance These were determined for two bodies and three bodies. In order to measure the force experienced by a follower at a stable position, one bearing was fixed to the rod, with a pulley system of negligible mass and friction attached to the rod at the top, as shown in Figure. Figure : The design of the pulley system used to measure the forces on a follower near its stable configuration. Fixing the bearing to the axle would cause the axle to rotate with the follower allowing the pulley to either wind up, if the follower was being pulled away from the swimmer in front of it, or wind down, if it was being pulled toward the swimmer in front of it. This works to simplify the force model greatly since friction is negligible on the bearings and at the stable configurations, the net force is zero so the only force that matters is the force used to pull the follower towards the swimmer in front of it or away from it, which is just gravity. The rest of the bearings were allowed to rotate freely. Both for the two body and three bodies, only one swimmer was held fixed, as shown in figure 3. 3

4 Figure 3: Configurations for each problem. The number above each wing corresponds to number put on each wing as seen in Figure 1. The shaded wing denotes which one the force was applied to. 3 Mathematical Model When measuring the position of stable configurations in formation, it is important to do so in terms of the wavelength of the leading swimmer s path, λ. A quantity called the Schooling Number was defined S = G λ (1) in order to measure the gap in terms of wavelength []. The Schooling Number of a stable position was denoted S i = G i () λ A Schooling Number near integer value would be indicative of the paths of the two swimmers nearly coinciding and being in phase, while a Schooling Number near half-integer value would imply the two swimmer s paths were in anti-phase. Experimentally, the forces at each stable position G i are thought to obey Hooke s Law, acting as a spring, giving an approximation F k(g G i ) (3) with k being the spring constant and G being the new schooling number as a result of the perturbation from the pulley. This is due to the fact that if the follower is perturbed from S i, it always returns back to that position. 4 Results The results obtained from the two-body problem for force measurements at the first stable schooling number, S 1 and the second stable schooling number, S are shown in Figure 3. The Schooling Numbers for S 1 and S when no force is being applied were extremely close to integer values, implying that the gap distance at the stable 4

5 position is an integer value of the leader s wavelength. The data points were found to have fit remarkably well on the curve y(x) = e x c µ sin[π(x c)] (4) where c and µ are free parameters. This suggests that as swimmers move farther apart, the wake interaction felt by a follower at these stable configurations decays exponentially as a sine wave. Force Measurements on Follower in Two-Body Problem at S1 and S Force Data at S1 Force Data at S Figure 4: Data from the two-body problem for the two stable Schooling Numbers S 1 and S at amplitude A =.cm. The measurements taken for the three-body problem provided strong evidence for the initial hypothesis that the forces felt on a wing in a three body problem could be modeled similarly as they are in a two body interaction. The data for the force measurements in the two-body problem and both configurations of the three-body problem are shown in Figures 5-8. Again, the curve that fitted the points in Figure 3 fitted the points in these figures reasonably well. 5

6 Force Measurements for Follower at S Two Body Formation nd Position in Three Body Formation 3rd Position in Three Body Formation Figure 5: Data points from the two-body and three-body problem at amplitude A 1 = 1.5cm 3 Follower in Two Body Formation Follower at nd Position in Three Body Formation Follower at 3rd Position in Three Body Formation Figure 6: Data from Figure 5 in individual plots 6

7 Force Measurements for Follower at S1 Two Body Formation nd Position in Three Body Formation 3rd Position in Three Body Formation Figure 7: Data points from the two-body and three-body problem at amplitude A =.cm. 3 Follower in Two Body Formation Follower at nd position in Three Body Formation Follower at 3rd Position in Three Body Formation Figure 8: Data from Figure 7 in individual plots There are two remarkable results depicted in figures 5-8. The first result was that for each configuration of swimmers tested the Schooling Number of the follower when no force was applied was near integer value. The second result was how close not only the two data points from the Three Body Problem fit with the Two Body problem, but how well they match each other. This provides evidence that perhaps the forces enacted upon a follower in formation can be localized, which is to say that in three bodies, the only forces that matter for a follower are the ones that come from the swimmer directly in front of it. 7

8 5 Discussion The results obtained from the two-body and three-body problems were remarkably similar, implying that no matter where the follower that was experiencing the applied force was in the Three Body formation, it was as if that follower was only swimming behind one body. Another surprising find was how the Schooling Number for each configuration tested was close to an integer value, meaning the gap distance between two adjacent wings was close to an integer value of the leader s wavelength λ. This provides evidence that the stable positions of swimmers happens when their paths are in phase meaning that these sorts of configurations can happen when a follower attempts to trace out the same path as the leader in front of it. Given the results, the next step in the research would be to complete a force analysis on a follower in a four-body problem and five-body problem with the aim of generalizing the results in this experiment to an N-body problem, where N is an arbitrary integer. 6 Acknowledgements I would like to dedicate this section of my paper to everyone who has helped me with my research this Summer. I would first like to provide Joel W. Newbolt with much thanks for all of the help and guidance he has provided me through this process. Thank you to Pejman Sanaei for helping me with my literature review early on in my research. I would then like to thank my Advisor Leif Ristroph. Lastly, I would like to thank the Applied Math Lab for allowing me to use their equipment and lab space to run this experiment. References [1] Steven J Portugal, Tatjana Y Hubel, Johannes Fritz, Stefanie Heese, Daniela Trobe, Bernhard Voelkl, Stephen Hailes, Alan M Wilson, and James R Usherwood. Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature, 55(7483):399, 14. [] Alexander D Becker, Hassan Masoud, Joel W Newbolt, Michael Shelley, and Leif Ristroph. Hydrodynamic schooling of flapping swimmers. Nature communications, 6:8514, 15. [3] PBS Lissaman and Carl A Shollenberger. Formation flight of birds. Science, 168(3934):13 15, 197. [4] David Willis, Jaime Peraire, and Kenneth Breuer. A computational investigation of bio-inspired formation flight and ground effect. In 5th AIAA Applied Aerodynamics Conference, page 418, 7. 8

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