As a first step, let us make a short description of the C like fast start just to recall by a few snapshots (see Fig. 1 and the related animation reported in Movie S1) the main phases of this pretty elaborated maneuver which obeys to the conservation of both linear and angular momenta, since no external actions are applied. The fish willing to suddenly accelerate and change its swimming direction initiates a preparatory phase via a rotation of its tail which induces a simultaneous opposite rotation of the body fixed frame.
Snapshots of the C-start maneuver of a neutrally buoyant fish from the numerical simulation. The relative animation is reported in Movie S1.
The successive propulsive phase, corresponding to the rapid return of the tail to the position aligned with the forward axis, gives rise to a substantial velocity boost in the same direction while the whole motion is accompanied by a significant release of vorticity. The kinematic performance of the C-start maneuver for a neutrally buoyant fish may be furtherly appreciated by the velocity components reported in Fig. 2 where we see that during the preparatory phase, i.e. for (0 le t/T < 0.5) when the tail is raised towards the head (see Fig. 1), the body fixed frame starts to counter-rotate with an angular velocity (Omega) whose maximum occurs approximately for (t/T = 0.5). A relatively small forward velocity U from right to left (i.e. negative in sign) is also obtained halfway, but a much larger forward speed is finally achieved at the end of the propulsive phase when the tail is pushed back. No comments are made about the lateral velocity component V since, in a first approximation, its presence is quite negligible.
Velocity components for the C-start maneuver of a neutrally buoyant fish.
The literature on the subject was mostly focused on the study of the vortex shedding and of the vortical wake geometry as a potential source of comprehension, while a little attention was given to the added mass that we consider instead of primary importance for the maneuver. For a quantitative evaluation of all these contributions, we rely on the conservation of the linear impulse along the forward direction:
$$begin{aligned} (m + m_{11}),U = {{mathcal {P}}}_1 end{aligned}$$
(1)
where m is the body mass and (m_{11}) is the added mass coefficient as properly defined when deriving the full system of equations (12) reported in the Methods section. Namely, Eq. (1) represents the first equation of the system once all the contributions but the one containing the unknown forward velocity U are grouped together within a single term ({{mathcal {P}}}_1 = – {P_v}_1 – {P_{sh}}_1 – m_{12} V – m_{13} Omega) to ease the interpretation of the results. Specifically ({{mathcal {P}}}_1), beyond the component ({P_v}_1) associated to the shed vortices and ({P_{sh}}_1) associated to the shape deformation, includes the coupling terms given by the lateral and angular velocities times the proper added mass coefficients (m_{12}) and (m_{13}), respectively. We may easily appreciate from Fig. 3a the very large difference between ({{mathcal {P}}}_1) and its vortical component, obviously covered by the left aside terms whose large impact on the maneuver clearly appears. It is interesting to evaluate the relative weight of the various terms to highlight the overwhelming predominance of the coupling term (-m_{13}Omega) which provides a substantial momentum transfer from the angular to the forward direction (diagrams reported in Fig. S2a). All the terms covering the above difference are shown to become negligible at the end of the propulsive phase where the deformation is over and the fish returns to its straight configuration. Actually, in this condition the total and vortical impulses ({{mathcal {P}}}_1) and ({-P_v}_1) perfectly coincide, hence we may assess that the value of the final swimming velocity at the end of the C-start maneuver may be obtained by accounting only for the shed vortices contribution15. At the same time, the vortical wake is shown to be unable to give a correct picture of the global physical phenomenon since all the other terms, in a way related to the added mass, have a dominant influence during the transient phase. By following the same reasoning, let us write the equation for the angular momentum:
$$begin{aligned} (I_{zz} + m_{33}) Omega = {Pi } end{aligned}$$
(2)
where, as before, the term ({Pi }) is grouping together all the other contributions but the one containing the angular velocity (Omega), while (I_{zz}) is the body moment of inertia and (m_{33}) is the proper added mass coefficient. Analogously, the difference between ({Pi }) and its vortical contribution ({-Pi _v}), reported in Fig. 3b, shows again the relevance of the left aside terms on the maneuver with a special regard to the coupling ones due to added mass (reported in Fig. S2c). At this point, since we have verified the limited role of the vortical wake for understanding the C-start, we may now pass to the dynamics of the maneuver to account for the effects of the added mass variability. Namely, by taking the time derivative of Eq. ((1)), we obtain
$$begin{aligned} frac{dU}{dt} = frac{1}{m + m_{11}}frac{d{{mathcal {P}}}_1}{dt} -frac{U}{m + m_{11}}frac{dm_{11}}{dt} end{aligned}$$
(3)
where the acceleration (frac{dU}{dt}) is split into two forcing terms. The first one depends directly on the time derivative of the forward impulse ({{mathcal {P}}}_1), while the second one depends on the time derivative of the added mass coefficient (m_{11}) along the forward direction. Both terms on the right hand side of Eq. ((3)) are divided by the sum of the body mass and of its added mass coefficient (m_{11}). Hence, the added mass coefficient accounting for all the water set in motion by the body forward translation behaves like the body mass, i.e. the smaller its value, the more effective are the forcing terms on the body acceleration. Moreover, the time derivative of the added mass coefficient (m_{11}) appears also as a forcing term which, for a reducing value of (m_{11}), may provide a boost in the body forward velocity, as highlighted by Spagnolie and Shelley16.
Fluid impulses for C-start maneuver: (a) total forward impulse ({{mathcal {P}}}_1) and its vortical contribution ({P_v}_1); (b) total angular impulse ({Pi }) and its vortical contribution ({Pi _v}).
By proceeding in an analogous way, similar equations may be obtained for the lateral and angular velocity components but, since the lateral velocity is much smaller and less important than the angular one, we report here only the expression for the angular acceleration:
$$begin{aligned} frac{dOmega }{dt} = frac{1}{I_{zz} + m_{33}}frac{dPi }{dt} -frac{Omega }{I_{zz} + m_{33}}frac{dm_{33}}{dt} end{aligned}$$
(4)
where the first term on the right hand side depends on the time derivative of the angular impulse (Pi), while the second one accounts for the variation of the added mass coefficient (m_{33}). For an easier understanding of the effects due to the added mass variation on the forward and the angular acceleration experienced by the fish, we reported in Figs. 4a and 5a, respectively, the time history of the added mass coefficients (m_{11}) and (m_{33}) while the behaviour of all the other coefficients is reported for completeness in Fig. S3. The total forward and angular accelerations and their contributions as given by Eqs. (3) and (4) are reported in Figs. 4b and 5b. In the first one, i.e. Fig. 4b, we observe how the two combined contributions always give rise to an acceleration from right to left (with a negative sign in our frame of reference) until the end of the maneuver. To this regard, according also to Fig. 4a, the time history of the second term on the r.h.s. of Eq. ((3)), accounting for the added mass variation, represents the main source of acceleration in the forward direction along the propulsive phase, even though a lighter opposite acceleration, substantially a drag, is shown during the preparatory phase. At the same time, the term accounting for (frac{d{{mathcal {P}}}_1}{dt}) shows a quite similar but opposite behaviour since the favorable effect appears during the preparatory phase, while the resistive effect occurs during the propulsive phase. By looking at the different components of (frac{d{{mathcal {P}}}_1}{dt}) reported in the Supplementary Material (see Fig. S4a), we have a further assessment of the dominant role played by the coupling among the angular and the forward velocities. With regard to the angular acceleration reported in Fig. 5b, the term related to the variation of the added mass coefficient (m_{33}) goes along with the time derivative of the angular impulse (dPi /dt) for most of the entire maneuver. The cooperative action of these two terms enhances the fish capability to perform quick turnings leading to a large angular velocity (Omega) which also has a favourable influence on the forward velocity through the coupling terms included in ({{mathcal {P}}}_1) as reported for completeness in Fig. S4b.
Time history of (a) the added mass coefficient (m_{11}) and of (b) the forward acceleration contributions for the C-start maneuver.
Time history of (a) the added mass coefficient (m_{33}) and of (b) the angular acceleration contributions for the C-start maneuver.
The presence of an undulatory motion cooperating with the main C-shape bending fully maintains the validity of the above reasoning about the relevance of the added mass for a good maneuverability. Indeed, the addition of a proper traveling wave is even enhancing the full deformation by leading, on the one side, to larger values of the added mass coefficients together with their time variation and, on the other side, to an increase of the angular velocity, which keeps providing the predominant forward momentum transfer. The increased deformation involving a larger amount of water to be accelerated was also mentioned by Gazzola et al.17 as a fostering effect for the C-start performance. The snapshots in Fig. 6 and the related animation in Movie S2 give a first glance evaluation of the more efficient maneuver, while the diagrams in Fig. 7 show the larger forward and angular velocities compared with the ones without traveling wave. Further figures on this case, quite similar to the previous ones for the basic C-start, are reported in Figs. S5 and S6.
Snapshots of the C-start maneuver combined with a wave undulation from the numerical simulation. The relative animation is reported in the Movie S2.
Comparison between forward and angular velocity components for the C-start maneuver with and without wave undulation.
