Two-dimensionally stable self-organization arises in simple schooling swimmers through hydrodynamic interactions
DD R A F T Two-dimensionally stable self-organization arisesin simple schooling swimmers throughhydrodynamic interactions
Melike Kurt a , Amin Mivehchi b , and Keith W. Moored b,1 a Aerodynamics and Flight Mechanics Group, Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, UK; b Mechanical Engineeringand Mechanics, Lehigh University, Bethlehem, PA, 18015, USAThis manuscript was compiled on February 9, 2021
We present new experiments and free-swimming simulations of apair of pitching hydrofoils interacting in a simple school. The hy-drofoils have an out-of-phase synchronization and their arrangementis varied from in-line to side-by-side arrangements through a seriesof staggered arrangements representing the two-dimensional inter-action plane. It is discovered that there is a two-dimensionally sta-ble equilibrium point for a side-by-side arrangement. In fact, thisarrangement is super-stable meaning that hydrodynamic forces willpassively maintain this arrangement even under external perturba-tions and the school as a whole has no net forces acting on, driftingit to one side or the other. Moreover, previously discovered one-dimensionally stable equilibria driven by wake vortex interactionsare shown to be, in fact, two-dimensionally unstable , at least for anout-of-phase synchronization. Additionally, the stable equilibrium ar-rangement is verified for freely-swimming foils undergoing dynamicrecoil motions. When constrained, the swimmers experience a col-lective thrust and efficiency increase up to 100% and 40%, respec-tively, in a side-by-side arrangement. However, in a staggered ar-rangement where there is direct vortex impingement on a follower,an even higher efficiency improvement of 87% is observed, which iscoupled with a 94% increase in the thrust. For freely-swimming foils,the recoil motion attenuates the performance improvements showinga more modest speed and efficiency enhancement of up to 9% and6%, respectively, when the swimmers are at their stable equilibrium.These newfound schooling performance and stability characteristicssuggest that fluid-mediated equilibria may play a role in the controlstrategies of schooling fish and fish-inspired robots. collective locomotion | hydrodynamic interactions | fish schooling | pattern formation | collective performance S elf-organization of living systems is one of Nature’s mostubiquitous and mesmerizing phenomena. It arises across awide range of spatial and temporal scales from the cells in ourbodies (1) and swarming of microorganisms (2) to the flockingof birds (3) and schooling of fish (4). For macroscopic flyersand swimmers, a wide range of hypotheses have attributedcollective behavior to social interactions (5), protection againstpredators (6), food prospect optimization (7), and/or energeticbenefits (4). Our knowledge of the latter hypothesis is limitedsince it is regulated by complex hydrodynamic interactions.Yet, both the spatial organization (3, 8) and temporal syn-chronization (9–11) have emerged as major drivers of the hy-drodynamic interactions, and, consequently, the energetic costof locomotion and traveling speed of individuals in a collective.Still, our understanding of the force production and energeticsof schooling swimmers is mostly limited to canonical spatialarrangements such as a leader-follower in-line arrangement (12–15) and a side-by-side arrangement (11, 16–18), while there are fewer studies of staggered arrangements (19–22).Because of these studies it is commonly presumed that thespatial organization observed in schools is driven by the interestto maximize swimming efficiency or force production. However,another explanation was first proposed by Sir James Lighthill(23). The so-called Lighthill conjecture (24) postulates that thearrangements of fish in a school may be due to the interactionforces that push and pull the swimmers into a particular stableformation, much like the atoms in a crystal lattice. Indeed, thisidea of passive self-organization has shown promise in recentstudies where one-dimensional streamwise stability has beenobserved in pairs of in-line self-propelled foils (24, 25) or insmall schools of various arrangements (20), as well as in pairsof in-line hydrofoils with differing kinematics (26). While thesestudies have shown seminal results supporting the Lighthillconjecture, they have only probed the one-dimensional stabilityof arrangements. However, two-dimensionally or even three-dimensionally stable arrangements are required for the passiveself-organization of schools that produce two-dimensional orthree-dimensional flows.Here, we advance our understanding of the hydrodynamicinteractions of schooling swimmers in two ways. For the firsttime, we measure the two-dimensional stability of schoolingarrangements, which takes us closer to understanding therole of the Lighthill conjecture in schooling formations. We Significance Statement
Fish schools are fascinating examples of self-organization innature. They serve many purposes from enhanced foraging,and protection to improved socialization and migration. How-ever, our understanding of the hydrodynamic interactions inschools is primitive. It has been postulated that these interac-tions can regulate energy usage and speed, as well as pushand pull individuals thereby altering the school’s structure andfunction. We have discovered that stable arrangements ofswimmers can arise in two-dimensional schools through pas-sive hydrodynamic forces alone. In these stable arrangements,swimmers also experience speed and efficiency benefits. Thisopens the door to considering that the structure and function offish schools may be more strongly regulated by hydrodynamicinteractions than previously known.
M. K. helped design the study, gathered and processed experimental measurements, and draftedthe manuscript. A. M. helped design the study, gathered and processed the numerical data, andhelped revise the manuscript. K. M. helped design the study, and helped revise the manuscript.The authors declare no conflict of interest. To whom correspondence should be addressed. E-mail: [email protected]
February 9, 2021 | vol. XXX | no. XX | a r X i v : . [ phy s i c s . f l u - dyn ] F e b R A F T Fig. 1.
Schematic of positions of the follower hydrofoil relative to the leader. Shown are two coarse rectangular grids of . c spacing ranging from − . ≤ X ∗ ≤ , . ≤ Y ∗ ≤ . , and . ≤ X ∗ ≤ , − . ≤ Y ∗ ≤ . , as well as a refined rectangular grid of . c spacing ranging from . ≤ X ∗ ≤ . , − . ≤ Y ∗ ≤ . .In total there are 180 grid points. discover that many of the one-dimensionally stable formationspreviously observed are, in fact, unstable once the cross-streamstability is considered. Yet, we still find that a side-by-sidearrangement is two-dimensionally stable providing supportfor the hypothesis that this arrangement observed in realfish (11) may be due to passive self-organization. Second, wemeasure the force production and energetics of two interactinghydrofoils throughout a plane of possible arrangements rangingfrom in-line to side-by-side by passing through the possiblestaggered arrangements. We reveal that there is a thrust andefficiency optimum in a slightly-staggered arrangement wherethere is direct vortex impingement on the follower. Experimental Approach and Results
To examine the flow interactions that occur in schools, fullswimmer models can be readily used in numerical studies (19),however, these models are difficult to implement experimen-tally. Instead, experiments typically use oscillating hydrofoilsas a simple model of the propulsive appendages of animals(12–14, 16, 17, 22, 24–27). Importantly, these oscillating hy-drofoils capture the salient unsteady fluid mechanics of theadded mass forces, circulatory forces, and shed vortices.Following this simple model approach, experiments wereconducted for two pitching hydrofoils immersed in a waterchannel. The flow over the hydrofoils was restricted to benominally two-dimensional. The arrangement of the hydro-foils was varied by manipulating the follower hydrofoil posi-tion in the streamwise and cross-stream directions as shownin Figure 1. The dimensionless distances were normalizedby the chord length as X ∗ = x/c , and Y ∗ = y/c . Theleader hydrofoil was prescribed a sinusoidal pitching motion of θ L ( t ) = θ sin(2 πft ), where the oscillation frequency is f , andthe pitching amplitude is θ , which is related to the peak-to-peak trailing edge amplitude as A = 2 c sin( θ ). The followerwas pitched similarly as θ F ( t ) = θ sin(2 πft + φ ) with a fixedphase difference or synchrony of φ = π throughout the study.Moreover, the oscillation frequency and the dimensionless am-plitude, A ∗ = A/c , were also fixed throughout the study at f = 0 .
98 Hz and A ∗ = 0 .
25, which gives a fixed reducedfrequency of k = fc/U = 1, and a fixed Strouhal number of St = fA/U = 0 .
25. These dimensionless numbers are typical of efficient biological swimming (28, 29). Direct force mea-surements were taken from each hydrofoil at every ( X ∗ , Y ∗ )position, which corresponds to a total of 180 arrangementsin the x - y plane. For further information about the waterchannel setup, actuator mechanisms, sensors, the methodsused, and the definition of the performance coefficients foran individual hydrofoil as well as the collective, please see Materials and Methods and
Supplementary Information . Follower Force Map.
In order to probe the Lighthill conjecturein two-dimensions, the relative forces acting on the follower inthe ( x - y ) interaction plane must be examined. This is doneby constructing a force map , which is described below.Consider a frame of reference attached to the leader as inFigure 1. The relative lift, ∆ L , in the cross-stream directionand the relative thrust, ∆ T , in the streamwise direction aredefined simply as a difference between the forces acting onthe two hydrofoils as ∆ T = T F − T L and ∆ L = L F − L L ,where forces acting on the leader and follower hydrofoils aredenoted with ( . ) L and ( . ) F , respectively. Figure 2A and 2Bshow the relative force conditions that leads to the followereither moving towards or moving away from the leader in thestreamwise ( x ) and cross-stream ( y ) directions.First, consider the relative lift for the positive x - y plane.The follower is pushed away from the leader in the cross-streamdirection ( l ) when the relative lift force is greater than zero(Figure 2A). This condition arises either when lift forces actingon the foils are in the same direction and L F > L L , or whenthey are acting in opposite directions and pointing away fromeach other ( L L < ↓ , L F > ↑ ). In contrast, the follower ispulled towards the leader when the lift forces are acting in thesame direction and L L > L F or acting in opposite directionsand pointing towards each other ( L L > ↑ , L F < ↓ ).Next, consider the relative thrust force in the positive x - y plane. A positive relative thrust force (∆ T >
0) acts to movethe follower towards the leader, which arises when T F > T L .In contrast, when T L > T F the relative thrust force is negative(∆ T <
0) and the follower moves away from the leader in thestreamwise direction as shown in Figure 2B. If T L = T F thenthe leader and follower swim at the same speed and do notmove closer or apart.To visualize the directions of the relative forces acting on the et al. R A F T Fig. 2.
Typical conditions leading to positive and negative relative (A) lift and (B) thrust. (C) Follower force map with an out-of-phase synchrony between the leader and follower,i.e. φ = π . The arrows on the force lines indicate the direction that the follower would move in relative to the leader if it were free-swimming. follower throughout the x - y plane we constructed a force map,which is a novel visualization made up of force lines (Figure2C). Put simply, the force map conveys the direction that thefollower would move in as observed by the leader. The forcemap is constructed with the origin located at the leading edgeof the leader and a relative force vector ( F rel = − ∆ T ˆx + ∆ L ˆy )is determined at each of the measurement positions detailed inFigure 1, such that a relative force vector field is created. Forcelines are then graphed as lines that are everywhere tangent tothe local relative force vector field, analogous to streamlines. Observed Equilibria.
The force map reveals four critical pointsor equilibria where the relative force vector is equal to zero,that is, ∆ T = 0 and ∆ L = 0, which are marked by bluecircles in Figure 2C. The first equilibrium point is located at( X ∗ , Y ∗ ) = (0 , .
6) in Region 1 where the leader and followerare interacting in a side-by-side arrangement. Interestingly, asthe force lines merge at this point, their direction indicatesthat this critical point is a stable equilibrium point in two-dimensions. Therefore, when any perturbations move thefollower away from this point, forces will arise to return thefoil back to this location. Another critical point in Region 2is located at ( X ∗ , Y ∗ ) = (1 . ,
0) in the leader’s wake wherethe follower is directly in-line with the leader. Region 2represents an equilibrium point that is stable to streamwise perturbations, but unstable to cross-stream perturbations,that is, an unstable saddle point . In Region 3 there is anequilibrium point located at ( X ∗ , Y ∗ ) = (1 . , .
3) representinga staggered arrangement of the foils. This equilibrium pointrepresents an unstable source point where the force-lines spiraloutward from it. Finally, Region 4 reveals another unstablesaddle point for a staggered arrangement with the followerlocated at ( X ∗ , Y ∗ ) = (1 . , . Collective Performance.
Beyond probing the Lighthill conjec-ture in two dimensions, the collective performance was alsomeasured in the same interaction plane. Figure 3 presentsthe normalized collective thrust, C ∗ T,C , and power, C ∗ P,C , thecollective lift, C L,C , and the normalized collective efficiency, η ∗ C , as a contour map of ( X ∗ , Y ∗ ). The normalized per-formance metrics compare the collective performance of theleader-follower pair with that of two isolated hydrofoils. Notethat in contrast to the relative forces that were discussed in theprevious section, a collective lift of zero does not necessarilymean zero lift for the foils individually.Figure 3A shows the normalized collective thrust as a func-tion of the streamwise and cross-stream position of the followerrelative to the leader. Generally, the collective thrust stayswithin the range of 1 ≤ C ∗ T,C ≤
2, which indicates that thecollective performs better than two hydrofoils in isolation for
Kurt et al.
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February 9, 2021 | vol. XXX | no. XX | R A F T Fig. 3.
Contour maps of normalized collective (A) thrust, and (B) power, (C) collective lift, and (D) normalized collective efficiency. The black dashed lines show the locationscorresponding to X ∗ = 0 , Y ∗ = 0 , and Y ∗ = 0 . . The leader position, size, and amplitude of motion is shown for reference. nearly the entire interaction plane considered here. The peaksin collective thrust can be grouped into two regions. First, isa region enclosing the side-by-side arrangements, along theline of X ∗ = 0 (black dashed line), where the collective isfound to achieve 40 − ≤ Y ∗ ≤ .
3. For the in-line arrangements, the col-lective thrust reaches a 74% peak increase over the hydrofoilsin isolation, whereas, the thrust reaches a 94% peak increasefor slightly-staggered arrangements along the Y ∗ = 0 .
125 line.This line is where direct wake vortex impingement onto thefollower is anticipated.Figure 3B presents the normalized collective power in theinteraction plane. For near wake interactions where the col-lective is in in-line or staggered arrangements ( X ∗ > . Y ∗ ), the collective power exhibits little variation from theisolated case, whereas, the side-by-side arrangements result inup to a 50% increase in power.Similarly, the collective lift coefficient is presented in Figure3C. For side-by-side arrangements, where a stable equilibriumpoint is located, the collective lift generation is found to benegligible. This means that when two individuals are swim-ming in this stable arrangement, the arrangement is, in fact, super-stable , that is, the relative distances between the swim-mers do not change and the pair of swimmers will remainswimming forward without a collective drift to one side oranother. Likewise, for the in-line interaction region, where anunstable equilibrium point is located, collective lift generationis found to be negligible as well. In Figure 3D, the collective efficiency is observed to increaseby 10 −
40% over that of isolated foils for the side-by-sideinteraction region. In the in-line interaction region, evenhigher peak efficiency increases are identified with up to a 74%increase for in-line interactions and an 87% increase for slightly-staggered arrangements along the Y ∗ = 0 .
125 dashed line. Itis clear that while both the side-by-side and in-line interactionregions see comparable thrust increases, the efficiency increasein the side-by-side interaction is tempered by a concurrentrise in power, whereas the efficiency increase in the in-lineinteraction region is solely driven by the increase in thrust.
Dynamic Schooling Interactions.
Within the interaction planeonly one two-dimensionally stable equilibrium point has beendiscovered for a side-by-side arrangement. However, this resultneglects the effect of dynamic recoil motions of freely swim-ming bodies, which may alter the physics of these interactions.In order to explore the effect of these dynamic recoil motionson the stability of the side-by-side equilibrium point and todetermine the free-swimming collective performance, two hy-drofoils free to move in both the streamwise and cross-streamdirections were simulated using a potential-flow-based numer-ical approach. For more details on the numerical approach,please see
Numerical Methods .Like the experiments, the simulated foils were prescribedsinusoidal pitching motions about their leading edge withan out-of-phase synchrony ( φ = π ). Each foil is assignedwith a mass normalized by their characteristic added mass, m ∗ = m/ ( ρsc ) = 2 .
68, which is comparable to biology (30)and matches previous work (27). Note that the constrained foil et al. R A F T measurements presented earlier have an effective dimensionlessmass of m ∗ = ∞ , since they do not exhibit recoil motions. Thehydrofoils have a virtual body that is not in the computationaldomain, but defines parameters to apply a drag force to thehydrofoils following a U high Reynolds number drag law (31).In free-swimming, the Strouhal number and reduced fre-quency are dependent variables and are outputs of the simu-lations, while the Lighthill number (see Numerical Methods )and dimensionless amplitude are input variables. By fixingthe Li = 0 . . ≤ A ∗ ≤ .
5, the Strouhal number is nearly constantat St ≈ . U scales with the amplitude and the pitch-ing frequency as U ∝ fA (31–33) and the reduced frequencyvaries over the range 0 . ≥ k ≥ .
62. Table 1 provides furtherdetails of the input/output variables for the simulations.Four cases with varying dimensionless amplitude are consid-ered. Each case was simulated with a range of initial positionsfor the follower within the region of − . ≤ X ∗ ≤ . . ≤ Y ∗ ≤ .
6, with 0 . Supplemen-tary Material for all of the simulation data). This reveals thata side-by-side arrangement is indeed a two-dimensionally sta-ble arrangement even for freely-swimming foils with dynamicrecoil motions.The simulation data is summarized in Table 1. Note thatduring self-propelled swimming the dynamic pressure basednormalization of the thrust coefficient is equal to the Lighthillnumber, a fixed quantity in our simulations, so the thrust andpower coefficients in the table are normalized by the addedmass forces instead (see
Numerical Methods ). The data showthat the equilibrium arrangement for each case spreads outwhen the reduced frequency decreases. This is the same trendobserved previously for a pitching foil in ground effect (27).The swimming speed and the thrust generation are increasedby 4 −
8% and 9 − − −
5% over a foil in isolation. The thrust and efficiency benefitsare observed to be tempered due to the dynamic recoil motionsof the foils.
Discussion and Conclusions
For the first time, we have discovered that a side-by-sidearrangement of pitching foils is not only two-dimensionally stable, but also two-dimensionally super-stable where the rela-tive distances between swimmers is constant and the school asa whole does not have forces acting on it to drift to one sideor the other during locomotion. Indeed, the school is shownto be super-stable for freely-swimming foils, which also enjoymodest speed and efficiency gains over swimming in isolation.This provides new evidence that the Lighthill conjecture (23)may play a role in school formation and spatial patterns ofswimmers even if only in a statistical sense akin to birds in aV-formation (10). Indeed, it has been observed (11) that abovea critical swimming speed, tetra fish organize in a side-by-side
Table 1. Simulation input and output data for four cases of varyingamplitude. There are 35 initial positions ( X ∗ , Y ∗ ) within the regionof − . ≤ X ∗ ≤ . and . ≤ Y ∗ ≤ . with . chord incre-ments simulated for each case. This gives a total of 140 simulations.The equilibrium position is denoted as ( X ∗ eq , Y ∗ eq ) and the origin isdefined at the leading edge of the leader. Parameters Case I Case II Case III Case IV A ∗ St k ( X ∗ eq , Y ∗ eq ) (0, 0.7468) (0, 0.8325) (0, 0.9239) (0, 1.0630) u ∗ C ∗ T,a C ∗ P,a η ∗ C iso T,a C iso P,a u iso lineup (or phalanx formation as described in the study) andthat they enjoy some energetic benefit. Our findings supportthis previous work and provide a new viewpoint that perhapsthe self-organization of the tetra fish is passive in nature andnot an active control strategy by the fish.While this is a provocative result, there are many differencesbetween schooling pitching foils and schooling tetra fish thatmake it difficult to draw a direct conclusion about the fish.One important difference is that the fish are composed of abody and fins, with it’s caudal fin undergoing a combinedheaving and pitching motion. However, recent work has shownthat both purely heaving and purely pitching foils experienceone-dimensional stability in an in-line arrangement (15), soperhaps the difference in kinematics is not consequential forthe existence of stable equilibria. Still, further work shouldaim to examine the passive stability of truly fish-like swimmersto directly answer this question.Another complicating factor is that it has been arguedthat three-dimensional swimmers in an infinite school expe-rience breakdown of there shed wake vortices (34), therebydisrupting the coherent vortex-body interactions that drive thein-line and staggered arrangement interactions. The side-by-side interactions, though, are dominated by oscillating dipoleflowfields produce from motions of the nearby bodies, whichcannot breakdown like wake vortices. This may mean that astable side-by-side arrangement and its performance benefitswould hold even for three-dimensional swimmers and/or denseschools. Further work in this direction is also warranted to pro-vide some understanding of how the two-body two-dimensionalinteractions of the current study translate to many-bodied andthree-dimensional interactions.Previous work (24–26) has shown that in-line arrangementshave multiple one-dimensionally stable equilibria, while thecurrent results show that those equilibria, at least in the near-wake of the leader, are in fact unstable in the cross-streamdirection. However, this does not indicate that these one-dimensionally stable points are irrelevant, but instead this workhighlights that the degree of stability falls along a spectrum.For instance, a fish swimming in the wake of another fishmay only need to actively control its cross-stream position inorder to maintain a schooling arrangement, which requires less Kurt et al.
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February 9, 2021 | vol. XXX | no. XX | R A F T Fig. 4. (A) Trajectory of the follower with respect to the leader leading-edge coordinates over 100 cycles for case IV. The initial positions that were simulated are represented bythe gray boxes. The color scale is mapped to the dimensionless time. (B) The leader and follower positions as well as the wake flows are highlighted at dimensionless times of t/T = 1 , , , and . The period of oscillation is T . control effort than actively controlling two degrees of freedom,but more effort than controlling none. While higher degrees ofstability can relieve needs for control strategies for swimmersand lead to completely passive self-organization, lesser degreesof stability may more subtly sculpt the schooling patternsobserved in fish by influencing the trajectory manifolds or thestatistical positioning of swimmers. Beyond the translationalstability of arrangements, it is unclear whether the orientationof a swimmer will also be stable to perturbations since it isunstable at least for some synchronizations and arrangements(35).Interestingly, when the thrust and efficiency performanceare considered, the ideal arrangement for maximizing per-formance is not the super-stable side-by-side arrangement,though modest thrust and efficiency benefits can be reapedin this arrangement. The optimal thrust and efficiency per-formance occurs for a slightly staggered arrangement, whichgives rise to interesting questions about whether animals swimin energetically optimal arrangements through more attentivecontrol or in more stable arrangements with less performancebenefits.This study provides a rich understanding of the interplayof stability and performance in schooling pitching foils andthese findings reveal hypotheses for understanding biologicalschooling. These findings also provide insights that aid in thedesign of multi-finned or schools of bio-inspired machines. Materials and Methods
Experimental Facility and Actuation Apparatus.
Experiments wereconducted for a simple collective consisting of two hydrofoils im-mersed in a closed-loop water channel, oscillating under prescribedsinusoidal pitching motions about an axis located 8 . U = 0 .
093 m/s wasimposed, which gives a chord-length based Reynolds number of Re = 9 , c = 0 .
095 m, and a span length of s = 0 .
19 m, which gives an aspectratio of AR = 2. Force Measurements.
An ATI Nano43 six-axis force sensor was usedto measure the thrust, lift and pitching moment acting on eachhydrofoil. An optical encoder recorded the angular position infor-mation, which was then used to compute the angular velocity foreach hydrofoil. The total instantaneous power input was calculatedas P T ( t ) = M θ ˙ θ . Here, the inertial power was determined from thesame experiments conducted in air, and was subtracted from the to-tal power, P T ( t ), to calculate the instantaneous power input to thefluid, P ( t ). Force measurements were taken for 100 oscillation cyclesfrom the leader and follower, and each experiment was repeated 10times. The time-averaged values were calculated for each of thesetrials, and their mean from 10 trials was calculated to determinethe time-averaged total thrust, lift, and power. Net thrust wasdetermined by subtracting the profile drag acting on the hydrofoil(s)from the time-averaged total thrust as follows, T net = T total − D .The definitions of the net thrust, C T , lift, C L , and power, C P ,coefficients and efficiency, η , are given as follows for the individualhydrofoils: C T = T net12 ρU cs , C L = L ρU cs ,C P = P ρU cs , η = C T C P , [1]where ρ is the fluid density and s is the span-length of the hydrofoils.Here, we also report collective performance parameters, that is, theaverage performance from the leader and the follower. The collectiveforce and power coefficients as well as the collective efficiency aredenoted with a C subscript and they are defined as, C T,C = T L + T F ρU ∞ cs , C L,C = L L + L F ρU ∞ cs ,C P,C = P L + P F ρU ∞ cs , η C = C T,C C P,C . [2]Note that, here, the performance coefficients were defined withcombined propulsor area, that is 2 cs , cancelling the one-half in thedenominator. Collective thrust and power coefficients, and efficiencywere reported as normalized values with the corresponding isolatedhydrofoil performance metric for comparison, and defined as follows, C ∗ T = C T C iso T , C ∗ P = C P C iso P , η ∗ = ηη iso . [3] et al. R A F T Here, the collective performance metrics are compared with thecombined values of two isolated hydrofoils. The isolated net thrust,drag, power and efficiency are C iso T = 0 . ± . C iso D = 0 . ± . C iso P = 0 . ± . η iso = 0 . ± . Numerical Methods.
To model the flow over a foil unconstrainedboth in the streamwise and the cross-stream direction, we useda two-dimensional boundary element method (BEM) based onpotential flow theory in which the flow is assumed to be irrotational,incompressible and inviscid. Previously, this method was usedto model flow over unsteady hydrofoils (31, 36, 37), and theirinteraction with a solid boundary and the associated performancefor constrained and unconstrained foils (17, 27). Further detailsabout the numerical method used here can be found in previouswork (27, 31).Since the hydrofoils are self-propelled a drag force is appliedto them, which follows a high Reynolds number U drag law (38).The drag force is determined by the properties of a virtual body,not present in the computational domain, which are determinedby the Lighthill number, Li . This dimensionless number is equalto Li = C D S wp , where C D is the drag coefficient of virtual bodyand S wp is a ratio of the wetted area of the virtual body andpropulsor to the planform area of the propulsor. This characterizeshow the virtual body drag coefficient and the body-to-propulsorsizing affects the balance of the thrust and drag forces. Its relationto the swimming speed was shown previously where more details ofthe virtual body can be found (31). In steady-state free-swimmingthe Li is equal to the dynamic pressure based thrust coefficient,that is Li = C T . To investigate the stability of the side-by-sideequilibrium point with recoil motions present, the Lighthill numberwas then set to Li = 0 .
3, which was the lowest value achievable forthe numerical stability of the current BEM formulation due to wakeimpingement on the foil bodies.In self-propelled swimming the dynamic pressure based thrustcoefficient simply equals the Lighthill number. This is a fixedquantity in the case of the simulations presented in the current studyregardless of whether the swimming speed increases or decreases.This means that the standard definition of the thrust coefficient ismeaningless in reflecting the thrust increase that leads to higherswimming speeds when two foils are freely swimming side-by-side.Thus, an added mass based thrust coefficient may be used as well asan added mass based power coefficient for consistency (31). Theyare defined as, C T,a = Tρf A cs , C P,a = Pρf A Ucs , η = C T,a C P,a . [4]The normalized added mass based thrust and power, and the nor-malized swimming speed are, C ∗ T,a = C T,a C iso T,a , C ∗ P,a = C P,a C iso P,a , u ∗ = uu iso , [5]respectively. ACKNOWLEDGMENTS.
This work was supported by the Na-tional Science Foundation under Program Director Dr. RonaldJoslin in Fluid Dynamics within CBET on NSF award number1653181 and NSF collaboration award number 1921809. Some ofthis work was also funded by the Office of Naval Research underProgram Director Dr. Robert Brizzolara on MURI grant numberN00014-08-1-0642.
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