School cohesion, speed, and efficiency are modulated by the swimmers flapping motion
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics School cohesion, speed, and efficiency aremodulated by the swimmers flapping motion
Sina Heydari and Eva Kanso † Aerospace and Mechanical Engineering, University of Southern California, Los Angeles,California 90089, USA
Fish schools are ubiquitous in marine life. Although flow interactions are thought tobe beneficial for schooling, their exact effects on the speed, energetics, and stability ofthe group remain elusive. Recent experiments suggest that flow interactions stabilizein-tandem formations of heaving foils. Here, we propose a minimal approach based onthe vortex sheet model that captures salient features of the flow interactions amongflapping swimmers, and we study the free swimming of a pair of in-line swimmers drivenwith identical heaving or pitching motions. We find that, independent of the flappingmode, the follower passively stabilizes at discrete locations in the wake of the leader,consistent with the heaving foil experiments, but pitching swimmers exhibit tighter andmore cohesive formations. Further, in comparison to swimming alone, pitching motionsincrease the energetic efficiency of the group while heaving motions result in a slightincrease in the swimming speed. These results recapitulate that flow interactions providea passive mechanism that promotes school cohesion, and provide novel insight into therole of the flapping mode in controlling the emergent properties of the school.
Key words:
Pattern formation, hydrodynamics, swimming, vortex-sheet model, heavingand pitching swimmers
1. Introduction
Fish schools are ubiquitous in aquatic life, with half of the known fish species thoughtto exhibit schooling behavior during some phase of their life cycle (Shaw 1978). However,the role of the fluid medium as a mediator of the physical interactions between swimmingfish remains unclear (Partridge & Pitcher 1979; Partridge 1982). Experimental evidencesuggests that fish modify their motions and reduce muscular effort when swimmingin vortex-laden flows (Liao et al. et al. et al. et al. et al. † Email address for correspondence: [email protected]
S. Heydari and E. Kanso leaderfollower single pressure forceleaderfollower single pitchingheaving (a)(b) pressure force
Figure 1.
A pair of swimmers undergoing (a) heaving motions at amplitude A h = 0 . A p = 15 ◦ . Snapshots of the velocity field (grey arrows) andfree vortex sheet of the leader (blue) and follower (red) are taken after steady-state swimmingis reached at a time instant where both swimmers are flapping downwards. Insets depict thepressure forces acting on each swimmer in the pairwise formation in comparison to a singleswimmer undergoing the same prescribed motion. formation (Tsang & Kanso 2013). These crystal lattice models do not capture that fishexhibit variable arrangements in field and laboratory experiments (Partridge & Pitcher1979; Marras et al. et al. et al. et al. et al. et al. et al. et al. et al. low-mediated schooling of heaving and pitching swimmers et al. et al. (2019) to assess the efficiency of lattice formations. High-fidelitycomputational fluid dynamics coupled to reinforcement learning algorithms were recentlyimplemented in pairwise interactions to optimize the flapping motion of the follower fishfor harnessing the wake of the leader (Verma et al. et al. et al. et al.
2. Problem formulation
A swimmer is modeled as a rigid plate of length 2 l , small thickness e ≪ l , andhomogenous density ρ , submerged in an unbounded, planar, fluid domain of density ρ f .The swimmer’s mass per unit depth is given by m = 2 ρel . An inertial frame ( e x , e y , e z )is introduced, such that ( e x , e y ) span the plane of motion. The vector x ≡ ( x, y ) denotesthe position of the geometric center of the swimmer in the ( e x , e y ) plane, and the angle θ its orientation relative to the e x -direction (see Appendix A and Figure A)The swimmer is free to move in the e x -direction under periodic heaving or pitchingmotions. Heaving consists of periodic lateral motions in the y -direction, of amplitude A h , at fixed angle θ = 0. Pitching refers to angular oscillations θ of amplitude A p ,with zero lateral motion y = 0 at the leading edge. The frequency of these heavingand pitching motions is denoted by f . Hereafter, we scale all parameter values using l as the characteristic length scale, 1 /f as the characteristic time scale, and ρ f l as thecharacteristic mass per unit depth. Accordingly, velocities are scaled by lf , forces by ρ f f l , moments by ρ f f l , and power by ρ f f l .In dimensionless form, the heaving and pitching motions are given byHeaving: y ( t ) = A h sin(2 πt ) , θ ( t ) = 0 , Pitching: θ ( t ) = A p sin(2 πt ) , y ( t ) = 0 . (2.1)The equation of motion governing the free swimming x ( t ) is given by Newton’s secondlaw m ¨ x = − F sin θ + S cos θ + D cos θ. (2.2)Here, the hydrodynamic forces acting on the swimmer consist of a leading edge suctionforce S , a pressure force F acting in the direction normal to the swimmer, and a skin dragforce D acting tangentially to the swimmer. The drag force D is introduced to emulatethe effect of fluid viscosity, while the hydrodynamic pressure force F is calculated in the S. Heydari and E. Kanso m ode l s l ope = . pressure dragdominant i n c r ea s i ng f r equen cy i n c r ea s i ng f r equen cy (c) (d) scaled speed experimental dataheaving airfoil(Ramananarivo et al. (2016)) heaving amplitude, A h (cm) s w i mm i ng s peed , U ( c m / s ) l og ( U ) −1 log( A h ) heaving pitching amplitude, A p º º º º º e x pe r i m en t s l ope = . (b) s w i mm i ng s peed , U ~ A p pitchingskin drag dominant ~ A p1.33 (a) heaving amplitude , A h heaving s w i mm i ng s peed , U U / ( f / ) Figure 2.
Swimming speed versus flapping amplitude for single swimmers. (a) Averageswimming speed at steady state for a heaving swimmer. (b) Average swimming speed atsteady state for a pitching swimmer. At small A p , skin drag is dominant and the speed scalessuper-linearly with A p . For A p > o , pressure drag is dominant and speed scales linearlywith A p . (c) Experimental data (black markers) of average swimming speed of a heaving foil(Ramananarivo et al. f / (yellow markers). (d) Comparing the swimming speed of our heaving swimmer model (bluecircles) to the frequency-scaled experimental data shown in (a) on a log-log scale. Both modeland experimental results scale super-linearly with heaving amplitude. context of the inviscid vortex sheet model. A detailed description of the method and itsnumerical implementation can be found in Nitsche & Krasny (1994); Huang et al. (2018),and a brief overview is given in Appendix A. Detailed expressions of the fluid forces andmoments acting on the swimmer are given in B.To assess the swimming performance, we use four metrics: the period-averaged swim-ming speed U = R t +1 t ˙ xdt at steady state, the thrust force T = S cos θ − F sin θ , the inputpower P required to maintain the prescribed heaving or pitching motions (see details inAppendix D), and the cost of transport defined as the input power P divided by theswimming speed U .
3. Single swimmers: numerical results and scaling analysis
We solve (2.2) in the case of a single swimmer and compute the period-averageswimming speed at steady state. In Figure 2(a) and (b), we show the steady state speedfor heaving and pitching swimmers, respectively, as a function of the flapping amplitude.In both cases, the speed increases monotonically, albeit that, when pitching, the increasescales differently at small amplitudes. To get insight into how the swimming speed U scales with the heaving and pitching amplitudes and frequency, it is instructive to use asimple scaling analysis.At steady state, the sum of forces acting on the swimmer is zero on average. Forheaving swimmers, the dominant forces are those due to the inertial added mass and low-mediated schooling of heaving and pitching swimmers ρ f (2 l )( A h f ) , where ρ f (2 l ) A h represents the mass of the laterally displaced fluid and A h f its acceleration.Skin drag scales as ρ f (2 l ) C f U , where C f ∼ p µ/ρ f (2 l ) U is the drag coefficient based onadapting Blasius theory to this inviscid fluid model (see Appendix C and White (1979)).Balancing inertia and drag, we arrive at ( A h f ) ∼ U / , which leads toHeaving: U ∼ ( A h f ) / . (3.1)The swimming speed scales super-linearly with the heaving amplitude and frequency.We test this scaling law in light of the experimental results of (Ramananarivo et al. f = 1 , , A p , the swimmer is almost parallel to the swimming direction,hence skin drag is dominant leading to the same scaling law as in the heaving case. Atlarge amplitude A p , pressure drag is dominant; it is well known that pressure drag scalesas U ; see, e.g., Moored & Quinn (2019). Balancing inertia and pressure drag, we arriveat U ∼ A p f . Put together, we havePitching: ( small A p : U ∼ ( A h f ) / , large A p : U ∼ A p f. (3.2)These scaling laws fit remarkably well the numerical results in Figure 2(b).
4. Pairwise formations: stability, speed, and energetics
We examine the steady state behavior of a pair of swimmers undergoing heaving andpitching motions while freely interacting via the fluid medium. In Figure 1, we showsnapshots of the flow field (grey arrows) and free vortex sheets in the case when the leader(blue) and follower (red) are heaving at A h = 0 . A p = 15 ◦ (Figure 1b). The snapshots are taken after the pair has reached steady state swimming inthe positive x -direction, and passively locked into a constant separation distance. At theseflapping amplitudes, the heaving swimmers experience longer transience and swim faster,whereas the pitching swimmers rapidly lock into a tighter formation (see SupplementalMovie 1).An analysis of the hydrodynamic pressure forces − F sin θ e x + F cos θ e y , where F is given in Appendix B, acting on each swimmer shows that compared to a singleswimmer, the distribution on the leader remains relatively unchanged. However, theforce distribution on the follower is affected by the wake of the leader, and the effect ismore pronounced for pitching swimmers; see insets in Figure 1(a) and (b). Specifically inthe pitching case, the follower experiences less resistance from the fluid, and a favorable S. Heydari and E. Kanso distance in wavelength, d h / λ t i m e , t distance in wavelength , d p / λ (a) δx f o r c e , po t en t i a l , (b) δx distance in wavelength, d h / λ distance in wavelength , d p / λ d h d p (c) (d) Figure 3.
Emergence of passive stable formations in a pair of heaving swimmers ( A h = 0 . A p = 15 ◦ ). (a) For heaving swimmers, the follower stabilizes atone of many discrete positions behind the leader where the gap (tail-to-head) distance d h isclose to integer multiple of of the wavelength λ = U/f of the leader motion. (b) For pitchingswimmers, the follower stabilizes at locations such that the tail-to-tail distance d p is close tointeger multiples of λ . Basins of attraction of each the first three equilibria are depicted ingradually more faint shades of grey. (c) and (d) Linear stability analysis: we perturb the positionof the follower about each of these equilibria and compute the total hydrodynamic force F x . Wesimultaneously sample data from the change in F x and perturbation strength δx , and plot δF x versus δx . Clearly, δF x acts as a restoring force. Taking the slope of δF x , we construct thehydrodynamic potential V on the follower. The potential well is deepest at the first equilibriumwhere the hydrodynamic interactions are strongest. force distribution (in the same direction of flapping) at the swimmer’s tail. At the instantshown in Figure 1(b), the downward flow due to the vortex sheet created by the leaderhelps the follower in its downward pitching motion.In Figure 3, we vary the initial separation distance between the two swimmers for theexamples shown in Figure 1. We find that for both heaving and pitching, the followertends to one of several discrete locations behind the leader at nearly digital values of d h /λ and d p /λ , respectively, where d h is the tail-to-head distance, d p the tail-to-tail distance,and λ = U/f the wavelength of the leader’s swimming trajectory; see Figure 3(top).Depending on initial conditions, the leader and follower reach one of these separationdistances and swim together in ordered formation. These findings are consistent with low-mediated schooling of heaving and pitching swimmers et al. d h and d p between the two swimmers; The basin of attraction ofeach relative equilibrium is highlighted in a different shade of grey in Figure 3(a,b).The pitching swimmers converge more rapidly to the corresponding equilibria, indicatingthat these equilibria are stronger attractors in pitching than in heaving. Further, thewavelength λ = U/f is smaller in pitching, and so is the actual separation distance atequilibria ( d p < d h ), indicating that pitching swimmers exhibit tighter formations.To quantitatively assess the linear stability of these equilibria, we perturb the positionof the follower about each equilibrium in the positive and negative x -direction with aninitial perturbation of size δx/l = 0 . δx and change in the total hydrodynamic force δF x = δ ( − F sin θ + S cos θ + D cos θ )acting on the follower in the x -direction. We scale the change in total force by U /l and the perturbation from equilibrium by d · /λ , where d · is either d h or d p . We samplesimultaneously the scaled change in total force δF x and scaled perturbation strength δx and we plot the results in the first row of Figure 3(c,d). The results are depicted in red △ markers for the first stable position, and in orange ◦ and yellow ✷ markers for the secondand third positions, respectively. Straight line fit for each of these data sets results instraight lines with negative slopes, implying that, for each of these equilibrium positions,the hydrodynamic force acts as a restoring force δF x = − Kδx that keeps the formationstable. Here, K is obtained numerically from the straight line fit. The value of K dependsmonotonically on the equilibrium position of the follower, with highest value at the firstequilibrium ( d h /λ ≈ d p /λ ≈ δF x = − ∂V /∂ ( δx ),where V = K ( δx ) / δx = 0. For both pitching and heaving, the formation is stable with weaker stability forlarger inter-swimmer distance. In the pitching formation the potential well is deeper (byabout 50%) for all equilibria, indicating faster convergence to the respective equilibrium.We next evaluate the advantages of these formations in terms of the speed andenergetics of the pair of swimmers in comparison to swimming alone. Figure 4 showsdetails of the time evolution at steady state of a single and pair of swimmers for the firstrelative equilibrium d h /λ ≈ d p /λ ≈ et al. S. Heydari and E. Kanso
20 21 22 23 24 2550 51 52 53 54 55 i npu t po w e r s w i mm i ng s peed t h r u s t f o r c e −1046810 time, t time, t c o s t o f t r an s po r t
40 41 42 43 44 45100 101 102 103 104 105 time, t time, t −100 (b)(a) pitchingheaving pitchingheaving Figure 4.
Instantaneous swimming performance (time-dependent speed, thrust, input power,and cost of transport versus time) for a single and pair of swimmers undergoing (a) heavingat A h = 0 . A p = 15 ◦ , respectively. Results are shown after the swimmershave reached steady state. From top to bottom, the swimming speed, thrust force, input powerand cost of transport are shown. Solid lines represent the instantaneous values and dashed linesrepresent time-period averages. swimmers have reached steady state. Specifically, we examine the range A h ∈ [0 , . A p ∈ [0 ◦ , ◦ ] for single swimmers and A h ∈ [0 . , .
7] and A p ∈ [10 ◦ , ◦ ] for pairs ofswimmers, where small amplitudes are ignored to ensure that hydrodynamic interactionsare sufficient for the spontaneous emergence of order formations. In pairwise interactions,we report all period-average values normalized by the corresponding values for a singleswimmer.When swimming alone, whether by heaving or pitching, an increase in the flappingamplitude monotonically increases the swimming speed, thrust, input power and costof transport; see left columns of Figure 5(a) and (b). Here, the swimming speed versusflapping amplitude for single swimmers is a reproduction of the results in Figure 2(a,b).Across all heaving amplitudes, the pairwise formation is about 5-10% faster than thatof a single heaving swimmer. Both the leader and follower experience an increase inthrust compared to the single swimmer, but require more power to swim in formationcompared to swimming alone, with extra power demand on the follower. The cost oftransport of the heaving formation is thus slightly higher (around 15%) compared toswimming alone. Thus, heaving swimmers slightly enhance their swimming speed whenin formation, albeit at a higher cost of transport.The formation of pitching swimmers is about 5% slower than swimming alone foralmost all flapping amplitudes. The leader experiences consistently lower thrust and thefollower consistently higher thrust compared to swimming alone. However, while thepower demand on the leader is comparable to the single swimmer, the power demandon the follower is significantly reduced for all amplitudes. Taken together, these resultslead to slightly higher cost of transport for the leader and significantly lower cost oftransport for the follower compared to swimming alone. Indeed, the cost of transport ofthe follower is a fraction of the single swimmer (around 25% at best), which in turn, causesthe formation to save a significant amount of power (around 35% at best) compared to low-mediated schooling of heaving and pitching swimmers heaving amplitude , A h pitching amplitude , A p º º º º (b) heaving amplitude , A h pitching amplitude , A p º º º º º (a) pitchingheaving pitchingheaving s w i mm i ng s peed i npu t po w e r t h r u s t f o r c e c o s t o f t r an s po r t s w i mm i ng s peed i npu t po w e r t h r u s t f o r c e c o s t o f t r an s po r t × 100 sc a l ed s peed sc a l ed po w e r sc a l ed t h r u s t sc a l ed c o s t sc a l ed s peed sc a l ed po w e r sc a l ed t h r u s t sc a l ed c o s t Figure 5.
Swimming performance (average speed, thrust, input power, and cost of transport)versus flapping amplitude for a single and pair of swimmers undergoing (a) heaving and (b)pitching motions, respectively. From top to bottom, average values of the swimming speed,thrust force, input power and cost of transport. Left columns (black markers) in (a) and (b)show the results for single swimmers. For the pair of swimmers, all of the results are scaled bythe corresponding quantity values for a single swimmer. The blue and red markers represent theresults for the follower and leader, respectively. The grey markers are the school average. swimming alone. These results imply that although the pairwise formation of pitchingswimmers experiences no enhancement in swimming speed compared to swimming alone,it reduces the cost of transport by a significant amount.
5. Conclusion
We analyzed the locomotion dynamics of actively flapping swimmers interacting pas-sively via the fluid medium in the context of the vortex sheet model. Within thetwo-swimmer model, we showed that hydrodynamic interactions lead to stable orderedformations, in which the follower falls into specific positions in the wake of the leader, andthe pair travels together at the same speed. This well-ordered ‘schooling’ behavior occursfor both heaving and pitching swimmers. Group cohesion is tighter and more stable forpitching swimmers. In heaving, the school swims slightly faster compared to swimmingalone, about 5-10% faster, albeit at a similar increases in cost of transport, especiallyfor the follower (about 20% higher cost for the follower and 15% for the formation).When pitching, the school swims at a slightly (about 5%) lower speed but has significant0
S. Heydari and E. Kanso energetic benefits, with up to 35% reduction in cost of transport for the formation andup to 75% for the follower.Our results are consistent with experimental findings of heaving foils (Becker et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. (2019), will be treated in futureworks.Acknowledgment. The authors would like to thank Michael J. Shelley and Leif Ristrophfor interesting conversations. This work is partially supported by the National ScienceFoundation grant CBET 15-12192, the Office of Naval Research grants 12707602 andN00014-17-1-2062, and the Army Research Office grant W911NF-16-1-0074.
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The coupled fluid-structure interaction between the swimming plate and the surround-ing fluid is simulated using an inviscid vortex sheet model. In viscous fluids, boundarylayer vorticity is formed along the sides of the swimmer, and it is swept away at theswimmer’s tail to form a shear layer that rolls up into vortices. In the vortex sheetmodel, the swimmer is approximated by a bound vortex sheet, denoted by l b , whosestrength ensures that no fluid flows through the rigid plate, and the separated shearlayer is approximated by a free regularized vortex sheet l w at the trailing edge of theswimmer. The total shed circulation Γ in the vortex sheet is determined so as to satisfythe Kutta condition at the trailing edge, which is given in terms of the tangential velocitycomponents above and below the bound sheet and ensures that the pressure jump acrossthe sheet vanishes at the trailing edge.To express these concepts mathematically, it is convenient to use the complex notation z = x + iy, where i = √− , y) denote the components of an arbitrary point inthe plane. The bound vortex sheet l b is described by its position z b ( s, t ) and strength γ ( s, t ), where s ∈ [ − l, l ] denotes the arc length along the sheet l b . The separated sheet low-mediated schooling of heaving and pitching swimmers Free vortex sheet Bound vortex sheet u + u − s = − l s = lγ ( s, t )Γ w e x e y z w ( s, t ) z b ( s, t ) F SD (a) (b) n θ
Figure 6. (a) Schematic of the vortex sheet model for a two-dimensional flapping swimmer.(b) Depiction of the different hydrodynamic forces acting on the swimmer. l w is described by its position z w ( Γ, t ), Γ ∈ [0 , Γ w ] where Γ is the Lagrangian circulationaround the portion of the separated sheet between its free end in the spiral center andthe point z w ( Γ, t ). The parameter Γ defines the vortex sheet strength γ = dΓ/ds .By linearity of the problem, the complex velocity w ( z, t ) = u ( z, t ) − i v ( z, t ) is asuperposition of the contributions due to the bound and free vortex sheets w ( z, t ) = w b ( z, t ) + w w ( z, t ) . (A 1)In practice, the free sheet l w is regularized using the vortex blob method to preventthe growth of the Kelvin-Helmholtz instability. The bound sheet l b is not regularized inorder to preserve the invertibility of the map between the sheet strength and the normalvelocity along the sheet. The velocity components w b ( z, t ) and w w ( z, t ) induced by thebound and free vortex sheets, respectively, are given by w b ( z, t ) = Z l − l K o ( z − z b ( s, t )) γ ( s, t ) ds, w w ( z, t ) = Z Γ w K δ ( z − z w ( Γ, t )) dΓ, (A 2)where K δ is the vortex blob kernel, with regularization parameter δ , K δ ( z ) = 12 π i z | z | + δ , z = x − i y (A 3)If z is a point on the bound sheet for which δ = 0, w b is to be computed in the principalvalue sense.The position of the bound vortex sheet z b is determined from the plate’s flapping( y ( t ) , θ ( t )) and swimming x ( t ) motions. The corresponding sheet strength γ ( s, t ) isdetermined by imposing the no penetration boundary condition on the plate, togetherwith conservation of total circulation. Let n ( s, t ) = − sin θ + i cos θ be the upward normalto the plate, the no penetration boundary condition is given byRe [ wn ] z b = Re [ w swimmer n ] , (A 4)where w swimmer = ˙ x − i ˙ y − i ˙ θ [¯ z b − ( x − i y )] . (A 5)Conservation of the fluid circulation implies that R l b γ ( s, t ) ds + Γ w ( t ) = 0.The circulation parameter Γ along the free vortex sheet z w ( Γ, t ) is determined by thecirculation shedding rates ˙ Γ w , according to the Kutta condition, which states that thefluid velocity at the trailing edge is finite and tangent to the flyer. The Kutta conditioncan be obtained from the Euler equations by enforcing that, at the trailing edge, thedifference in pressure across the swimmer is zero. To this end, we integrate the balanceof momentum equation for inviscid planar flow along a closed contour containing the4 S. Heydari and E. Kanso vortex sheet and trailing edge,[ p ] ∓ ( s ) = p − ( s ) − p + ( s ) = − dΓ ( s, t ) dt −
12 ( u − − u ) , (A 6)where Γ ( s, t ) = Γ w + R s − l γ ( s ′ , t ) ds ′ , − l s l , is the circulation within the contour and p ∓ ( s, t ) and u ∓ ( s, t ) denote the limiting pressure and tangential slip velocities on bothsides of the swimmer. Since the pressure difference across the free sheet is zero, it alsovanishes at the trailing edge by continuity, which implies that˙ Γ w = −
12 ( u − − u ) | s = − l . (A 7)The values of u − and u + are obtained from the average tangential velocity componentand from the velocity jump at the trailing edge, given by the sheet strength, evaluatedat s = − l u = u + + u − w − w swimmer ) n ] , u − − u + = γ. (A 8)Once shed, the vorticity in the free sheet moves with the flow. Thus the parameter Γ assigned to each particle z w ( Γ, t ) is the value of Γ w at the instant it is shed from thetrailing edge. The evolution of the free vortex sheet z w is obtained by advecting it intime with the fluid velocity, ˙¯ z w = w w ( z w , t ) + w b ( z w , t ) . (A 9) Appendix B. Forces and moments
The hydrodynamic force acting on the swimmer due to the pressure difference acrossthe swimmer is given by, Z l b n [ p ] ∓ ds = − F sin θ + i F cos θ, (B 1)where F = R l b [ p ] ∓ ds . The hydrodynamic moment acting on the swimmer about itsleading edge is given by M = Re (cid:20)Z l b i n ( z le − z b )[ p ] ∓ ds (cid:21) , (B 2)where z le is position of the leading edge s = ± l .It is known that the strength of the bound vortex sheet exhibits an inverse square rootsingularity at the edges (Saffman (1992); Eldredge (2019)). The singularity at the trailingedge is regularized by enforcing the Kutta condition as discussed above. To regularize thesingularity at the leading edge, we introduce a force parallel to the plate known as leadingedge suction (Eldredge 2019). Following the derivation provided in Eldredge (2019), wewrite the suction force, in dimensionless form as S = 2 πe i θ σ , (B 3)where σ is the suction parameter defined as σ = 12 ( ˙ y − l ˙ θ cos θ ) + Z l b γ ( s, t )2 πl Re (cid:16) ˜ z ( s, t ) + l ˜ z ( s, t ) − l (cid:17) / ds (B 4)where ˜ z ( s, t ) = z ( s, t ) − ze i θ , is the complex position of any vortex sheet present in thefluid written in the plate’s frame of reference. ˙ y − l ˙ θ cos θ is the velocity of the center of low-mediated schooling of heaving and pitching swimmers y -direction. Note that in equation B 3, the suction force is always positive(always a thrust force) and parallel to the plate.Note that the majority of the suction force is due to the vertical motion of the leadingedge relative to the surrounding fluid. For the pitching swimmer, since the leading edgehas no vertical motion, the contribution of the leading edge suction force to the totalthrust force of the swimmer is negligible. This is confirmed by our numerical experimentson a single pitching swimmer.Last, we introduce a drag force D that emulates the effect of skin friction due to fluidviscosity. This force is based on the Blasius laminar boundary layer theory as implementedby Fang (2016) in the context of the vortex sheet model. Blasius theory provides anempirical formula for skin friction on one side of a horizontal plate of length 2 l placedin fluid of density ρ f and uniform velocity U . In dimensional form, Blasius formula is D = − ρ f (2 l )( c f ) U , where the skin friction coefficient C f = 0 . / √ Re is given in termsof the Reynolds number Re = ρ f U (2 l ) /µ . Substituting back in the empirical formulaleads to D = − C d U / , where C d = 0 . p ρ f µ (2 l ). Following Fang (2016), we write amodified expression of the drag force for a swimming plate D = − C d ( U / + U / − ) , (B 5)where U ± are the spatially-averaged tangential fluid velocities on the upper and lowerside of the plate, respectively, relative to the swimming velocity U , U ± ( t ) = 12 l Z l − l u ± ( s, t ) ds − U. (B 6)We estimate C d to be approximately 0 .
02 in the experiments of Ramananarivo et al. (2016).
Appendix C. Numerical implementation
The bound vortex sheet is discretized by 2 n + 1 point vortices at z b ( t ) with strength ∆Γ = γ∆s . These vortices are located at Chebyshev points that cluster at the two ends ofthe swimmer. Their strength is determined by enforcing no penetration at the midpointsbetween the vortices, together with conservation of circulation. The free vortex sheetis discretized by regularized point vortices at z w ( t ), that is released from the trailikngat each timestep with circulation given by (A 7). The free point vortices move with thediscretized fluid velocity while the bound vortices move with the swimmer’s velocity.The discretization of equations (2.2) and (A 7, A 9) yields a coupled system of ordinarydifferential evolution equations for the swimmer’s position, the shed circulation, and thefree vorticity, that is integrated in time using the 4th order Runge-Kutta scheme. Thedetails of the shedding algorithm are given in Nitsche & Krasny (1994). The numericalvalues of the timestep ∆t , the number of bound vortices n , and the regularizationparameter δ are chosen so that the solution changes little under further refinement.Finally, to emulate the effect of viscosity, we allow the shed vortex sheets to decaygradually by dissipating each incremental point vortex after a finite time T diss from thetime it is shed into the fluid. Larger T diss implies that the vortices stay in the fluid forlonger times, mimicking the effect of lower fluid viscosity. For the results depicted inthis study, we used T diss ∈ [1 . , .
5] flapping period. We refer the reader to Huang et al. (2018) for a detailed analysis of the effect of dissipation time on the hydrodynamic forceson a stationary and moving plate in the vortex sheet model. Details of the numerical6
S. Heydari and E. Kanso validation in comparison to Jones (2003) and Jones (2005) are provided in Huang et al. (2016).
Appendix D. Swimming Energetics
Heaving motions are produced by an active heaving force F h acting by the swimmeron the fluid in the y -direction. The value of F h is obtained from the balance of linearmomentum on the swimmer in the y -direction,Heaving: m ¨ y = F y + F h . (D 1)Here, the hydrodynamic force F y acting on the swimmer in the y -direction is given by(B 1).Pitching motions are produced by an active moment M p acting by the swimmer on thefluid about the leading edge. The value of M p is obtained from the balance of angularmomentum about the swimmer’s leading edge,Pitching: I ¨ θ − Im[ m ( ˙ x + i ˙ y ) w l . e . ] = M + M p , (D 2)Here, I = m (2 l ) / w l . e . isthe swimmer’s velocity at the leading edge, and M is the hydrodynamic moment aboutthe leading edge given in (B 2).The power input by the swimmer into the fluid due to heaving and pitching motions,respectively, is given by Heaving: P h = F h ˙ y, Pitching: P p = M p ˙ θ.θ.