The performance of a flapping foil for a self-propelled fishlike body

Several fish species propel by oscillating the tail, while the remaining part of the body essentially contributes to the overall drag. Since in this case thrust and drag are in a way separable, most attention was focused on the study of propulsive efficiency for flapping foils under a prescribed stream. We claim here that the swimming performance should be evaluated, as for undulating fish whose drag and thrust are severely entangled, by turning to self-propelled locomotion to find the proper speed and the cost of transport for a given fishlike body. As a major finding, the minimum value of this quantity corresponds to a locomotion speed in a range markedly different from the one associated with the optimal efficiency of the propulsor. A large value of the feathering parameter characterizes the minimum cost of transport while the optimal efficiency is related to a large effective angle of attack. We adopt here a simple two-dimensional model for both inviscid and viscous flows to proof the above statements in the case of self-propelled axial swimming. We believe that such an easy approach gives a way for a direct extension to fully free swimming and to real-life configurations.

where the time derivative of the total impulse p p p gives the force acting on the body and ρ b is the body density. Let us now divide the whole body into an active part B T given by the tail and a completely passive one, named virtual body B V , whose presence is attested only by its mass and its viscous drag in the axial direction. By using a Cartesian frame of reference (e 1 , e 2 , e 3 ) and by isolating the unknown locomotion speed u 0 = U e 1 , the total motion of the entire body may be split into where u T is given by the prescribed heave and pitch motion of the tail where x 0 is the position of the pivot point and V and Ω are the lateral and angular heave and pitch velocity, respectively. Since "... the fish's muscular contractions can only determine changes in its shape relative to the centre of gravity." (Lighthill 70 1 ), the velocity of the virtual body u is taken to satisfy the conservation of linear and angular momenta for the entire body system, including the prescribed movement of the tail. By combining (S1) and (S2) we obtain The surface integrals appearing within the total impulse p may be decoupled into the contribution from the tail and that from the virtual body. By taking the component of (S4) along e 1 to solve for the locomotion along the axial direction, we assume the virtual body contribution to be represented by its overall resistance D, leading to where the axial component p of the impulse contains the contribution from the tail. By assuming zero initial conditions, (S5) gives: The scalar potential introduced by the Helmholtz decomposition is evaluated according to the related boundary conditions on the tail boundary ∂ φ ∂ n = u u u b · n n n T Finally, the potential impulse may be expressed in terms of the added mass coefficients introduced in the classical treatises (see e.g. 2 ) that, for completeness, are reported below. For a foil motion with unknown axial velocity U and prescribed lateral and angular velocity V and Ω respectively, we consider the Kirchhoff base potentials Φ 1 , Φ 2 and Φ 3 defined through the boundary conditions to have φ = UΦ 1 +V Φ 2 + ΩΦ 3 . It follows for the added mass coefficients in the axial direction m 1 j the expression The prescribed lateral and angular tail velocities within p φ , which are multiplied by m 12 and m 13 respectively, can be shifted to the r.h.s. to yield the equation for the axial body motion: The drag term appearing on the r.h.s. of (S9) is expressed as D = 1 2 ρU 2 LC D , where L is the body length and C D is the prescribed drag coefficient. The numerical solution of the equation, quite trivial at steady state, requires a simple numerical treatment to manage the transient phase of the locomotion velocity. As a final remark, the input power P is evaluated as where the lift is L = d p p p dt · e e e 2 and the moment is M = dπ π π dt · e e e 3 with the angular impulse about the tail leading edge defined as ρ |x| 2 (n n n × u u u + ) dS (S11) 2/6 2 Techniques and data for the simulations A flapping foil acting as the propulsor of a fishlike body has been studied by a well-known inviscid numerical procedure with the aim to suggest a neat and simple way to investigate the performance of oscillatory swimming fish. The flow solutions about the flapping airfoil are obtained by a potential code based on Hess and Smith 3 approach together with a suitable unsteady Kutta condition and a proper evolution of the wake behind the airfoil as indicated by Basu and Hancock 4 . Finally, the locomotion of the whole-body (airfoil + virtual body) is obtained by satisfying the conservation of total momentum in the forward direction. Further details on the adopted methodology can be found in Paniccia et al. 5 , where also lateral and angular directions are considered.
With regard to the viscous results reported in the manuscript, the flow solutions have been obtained by using a CFD solver for  Figure S1. Comparison between the current viscous model results and the ones by Lin et al. 8 for a self-propelled heaving and pitching foil.
the Navier-Stokes equations based on an immersed boundary method. The numerical code has been developed by Popinet 6 and it was already successfully adopted in the field of self-propelled fish locomotion by 7 among others. The forward locomotion of the whole-body follows from the evaluation of the axial force exerted by the surrounding fluid and from the solution of the Newton's second law in the axial direction only. The current viscous solver results are validated against the ones obtained by Lin et al. 8 for the same heaving and pitching conditions as shown in fig.S1. Let us now describe in more details the parameters we selected for the flapping foil and for the virtual body in front of it. We assume the body length L = 1 m and the tail length l, here taken as the reference length, equal to 1/7 L, ratio frequently observed in nature for real tuna. The presence of the virtual body is only attested by its mass m b and its resistance in terms of a prescribed drag coefficient C D . The values of m b is based on a NACA0018 airfoil geometry with chord length equal to 6l and unit density, i.e. m b ≈ 4.4 kg, and the value of C D ≈ 0.25 has been selected as the mean value of the experimental data by White et al. 9 for their robotic tuna. Finally, the flapping foil representing the tail is modeled as a NACA0012 airfoil of chord length l with mass m t equal to 0.08 kg, leading to a total mass of the whole-body m = m b + m t ≈ 4.5 kg. For further geometrical details, fig.S2 reports the sketch of the tail from the inset of fig.1 in the main text where the whole-body is shown. Figure S2. Sketch of the tail from the inset of fig.(1) in the main text.

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The heave motion of the tail is defined as where h 0 is the heave amplitude and ω is the oscillation angular frequency which is equal to 10π rad/s in the present case. The pitch motion is defined as where θ 0 is the pitch amplitude and the phase angle φ is equal to π/2. For pitch motions about the leading edge and for sufficiently small θ 0 it is possible to approximate the value of the non-dimensional peak-to-peak trailing edge amplitude with the following analytical expression (see also 10 ) where A h = 2h 0 /l is the non-dimensional amplitude for a pure heave motion. Two different trailing edge amplitudes have been considered, namely A T E = 1.0 and A T E = 1.5, for the ratio A h /A T E ranging within 0.4 ∼ 0.98 while the pitch amplitude θ 0 follows directly from (S14). For the sake of completeness, we summarize in Tab.S1 below all the input data used in our study.  Table S1. Values of h 0 l , A h A T E and θ 0 for A T E =1.0 (left) and A T E =1.5 (right).

The effect of forward oscillations
To proof that the oscillations in the forward velocity give a negligible contribution on large scale parameters like the cost of transport and the locomotion velocity we should realize a self-propelled motion with an axial velocity constrained to be perfectly constant, i.e. without the implicit oscillations. However, since the axial velocity is the unknown of the problem, we cannot make any constraint on this variable as we may do with the lateral and angular motions where it is quite easy to annihilate the values of the corresponding velocities (see 11 ). To overcome this conundrum we report here for comparison the results obtained by a self propelled approach and the ones obtained by the prescribed uniform stream. Figure (S3a) shows the forward velocity obtained by the self-propelled fishlike body for h 0 = 0.6 and A T E = 1.5. To compare this self-propelled case against an axial location fixed swimming case, we selected the mean forward velocity reached at steady state of the first case as the prescribed constant speed for the second one. It follows the same drag coefficient in both cases within the approximation obtained for the other variables. In fig.(S3b) we report the input power time evolution in one oscillation period for both the self-propelled and the fixed swimming cases. From the comparison we may appreciate a very small difference, less than 2%, between the two mean values that, in a first approximation, is quite negligible. This fact, explains why for the axial swimming, and exclusively in this case, each single result in one point of the parameter space may be indifferently obtained by the two mentioned approaches. Obviously, this is not true anymore for fully free swimming in presence of all the recoil motion components.
As an ultimate comment, when using the one or the other approach to explore the parameter space, the routes to find the optimal swimming performance are completely different. In fact, by taking the flapping foil data as a running parameter for the analysis, we may fix either the velocity, as in the prescribed stream approach, or the body drag coefficient, as in the self propulsion approach and this last procedure, in our opinion, is certainly preferable since the aim is to find these results for a certain fishlike body.