2. Impulse equation and self-propulsion

Here, we adopt the approach used in our previous works, where more details are reported (see e.g. Paniccia et al. Reference Paniccia, Graziani, Lugni and Piva2021a , Reference Paniccia, Graziani, Lugni and Piva2021b ), following the formulation presented in Kanso (Reference Kanso2009). Considering the fluid–body system with $V_f$ and $V_b$ the fluid and body volumes, respectively, as an isolated system with no external forces acting upon it, and denoting $\boldsymbol{F}_b$ as the force acting on the body and $\boldsymbol{F}_{\!f}$ as the force acting on the fluid, the following relation can be established:

(2.1) \begin{equation} \;\;\;\, \boldsymbol{F}_b + \boldsymbol{F}_{\!f} = 0, \end{equation}

which expresses the total momentum conservation in integral form

(2.2) \begin{equation} \frac{d}{{\rm d}t} \left ( \int _{V_b} \rho _b \, \boldsymbol{U}_b\, {\rm d}V + \int _{V_f} \rho \, \boldsymbol{U}{\rm d}V \right ) = 0, \end{equation}

where $\boldsymbol{U}_b, \rho _b$ are the velocity and density at any point in the homogeneous body while $\boldsymbol{U}, \rho$ refer to the same quantities in the surrounding fluid. By introducing the velocity of the body’s centroid, $\boldsymbol{U}_\textit{cm}$ , the first integral in (2.2) gives

(2.3) \begin{equation} \frac{d}{{\rm d}t} \int _{V_b} \rho _b \, \boldsymbol{U}_b\, {\rm d}V = \boldsymbol{F}_b = m_b\, \frac {{\rm d} \boldsymbol{U}_\textit{cm} }{{\rm d}t}, \end{equation}

where $m_b$ is the total mass of the body. The second integral in (2.2) represents the time variation of the fluid’s momentum and can be rewritten in terms of the hydrodynamic impulse $\boldsymbol{P}$

(2.4) \begin{equation} \frac{d}{{\rm d}t} \int _{V_f} \rho \, \boldsymbol{U}\, {\rm d}V = \boldsymbol{F}_{\!f} = \rho \, \frac {{\rm d} \boldsymbol{P}}{{\rm d}t}. \end{equation}

In a pure potential flow, the impulse $\boldsymbol{P}$ is confined to its irrotational (acyclic) component, while vorticity-related contributions are not considered. By assuming the normal outward from the body, it is defined in terms of the velocity potential $\phi$ as

(2.5) \begin{equation} \boldsymbol{P} = \boldsymbol{P}_\phi = - \int _{S_b} \phi \boldsymbol{n}{\rm d}S. \end{equation}

Equation (2.4) can be written as

(2.6) \begin{equation} \boldsymbol{F}_{\!f} = - \rho \frac{d}{{\rm d}t} \int _{S_b} \phi\boldsymbol{n}{\rm d}S, \end{equation}

which aligns with the formulation by Saffman (Reference Saffman1992). To analyse the motion of the body, we directly employ the impulse equation by combining (2.3) and (2.4) and by eliminating the time derivative (for a null initial condition), as also discussed in Kanso (Reference Kanso2009) and Paniccia et al. (Reference Paniccia, Graziani, Lugni and Piva2021a )

(2.7) \begin{equation} m_b\, \boldsymbol{U}_\textit{cm} = - \rho \boldsymbol{P}. \end{equation}

Now we split the impulse $\boldsymbol{P}$ into a part, $\boldsymbol{P}_{LOC}$ , due to the body locomotion and a part, $\boldsymbol{P}_\textit{SH}$ , due to the body shape deformation. The former can be expressed through the added-mass matrix $\boldsymbol{M}$ (see also Appendix A) yielding

(2.8) \begin{align} \rho \boldsymbol{P}_\textit{LOC} = \boldsymbol{M}\boldsymbol{U}_\textit{cm}, \end{align}

and (2.7) becomes

(2.9) \begin{equation} (m_b + \boldsymbol{M})\, \boldsymbol{U}_\textit{cm} = - \rho \boldsymbol{P}_\textit{SH}, \end{equation}

which shows that, upon enforcing a periodic shape deformation given by a non-reciprocal motion of the body (as defined in Purcell Reference Purcell1977), we may have locomotion in the absence of mean forces or accelerations.

As previously anticipated, to analyse the self-propulsion of a fish, we model the tail as a heaving and pitching plate attached to a virtual body that contains all the mass of the system while it does not experience either applied forces or moments. This implies that: (i) the virtual body does not contribute to the impulse $\boldsymbol{P}$ in (2.7); (ii) the only source of propulsion is the oscillating tail; and (iii) the centre of mass of the system is assumed to move solely with velocity $U_X$ in the ground reference frame, as a result of the tail’s oscillations. To this aim the tail undergoes prescribed heave oscillations in the $Y$ -direction and pitch oscillations with angular velocity $\varOmega (t) \boldsymbol{e}_3$ (positive counterclockwise) around the centre of rotation located at its leading edge (hereinafter denoted as LE or $\boldsymbol{x}_o$ ). As a result we will obtain the locomotion of the whole system.

The impulses $\boldsymbol{P} =\{{P}_X,{P}_Y\}$ appearing in (2.7) are evaluated in the ground frame. In the following, we will see that it is more convenient to use the components in the body-fixed frame $\{{P}_x,{P}_y\}$ , expressed by using the rotation matrix which maps the ground to the body frame, following the approach presented in Eldredge (Reference Eldredge2019) and Limacher (Reference Limacher2021)

(2.10) \begin{equation} R(\theta ) = \quad \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}, \end{equation}

the angle $\theta$ being assumed positive anti-clockwise. It follows that

(2.11) \begin{equation} \begin{bmatrix} P_x \\ P_y \end{bmatrix} = R(\theta ) \begin{bmatrix} P_X \\ P_Y \end{bmatrix} = \begin{bmatrix} P_X\,\cos \theta + P_Y\,\sin \theta \\ -P_X\,\sin \theta + P_Y\,\cos \theta \end{bmatrix}. \end{equation}

By using the rotation matrix, (2.7) becomes, in the body frame,

(2.12) \begin{equation} \left \{ \begin{array}{l} m_b\,u_x = - \rho \,P_x ,\\ m_b\,u_y = - \rho \,P_y, \end{array} \right . \end{equation}

where the body mass $m_b$ is estimated as reported in Appendix B.

Let us consider a flat plate whose half-length is $b=c/2$ undergoing imposed harmonic heave and pitch motions $(\dot h\boldsymbol{e}_2; \dot \theta \,\boldsymbol{e}_3 = \varOmega\boldsymbol{e}_3)$ , while its rotation centre moves with locomotion velocity ( $u_x, u_y$ ) expressed in the body frame. The outgoing normal to the upper face is $\boldsymbol{n} = \{-\sin \theta ; \cos \theta \} \simeq \{- \theta ; 1\}$ . The velocity normal to the plate is

(2.13) \begin{equation} U_2 = u_y + \dot h \,\cos \theta \simeq u_y + \dot h. \end{equation}

The relevant details to obtain the expression of the impulses in (2.12) are reported in Appendix A. By using (2.13), the impulse $P_y$ expressed according to (A11) becomes

(2.14) \begin{equation} P_y = \pi \,b^2 \left [ u_y + \dot h + b \dot \theta \right ]. \end{equation}

By combining both the first (2.12) and (A11), we obtain $u_x=0$ . Finally, from the second equation in (2.12) and (2.14), the solution $u_y$ may be obtained

(2.15) \begin{equation} u_y = \frac {- \pi \,\rho \,b^2}{m_b + \pi \,\rho \,b^2} \left [ \dot h + b \dot \theta \right ]. \end{equation}

The self-propulsion speed of the whole body is evaluated in the ground frame through the velocity $\boldsymbol{U}_\textit{cm} = \{U_X , U_Y \}$ which is expressed as

(2.16) \begin{equation} \begin{bmatrix} U_X \\ U_Y \end{bmatrix} = R(\theta ) ^T \begin{bmatrix} u_x \\ u_y \end{bmatrix} = \begin{bmatrix} \,\,\, \cos \theta \,u_x - \sin \theta \,u_y \\ \sin \theta \,u_x + \cos \theta \,u_y \end{bmatrix}. \end{equation}

The locomotion velocity in the body frame is given by $u_x=0$ and $u_y$ by (2.15), so (2.16) gives

(2.17) \begin{equation} \begin{array}{l} U_X= \;\;\, \dfrac {\pi \rho b^2}{\left(m_b + \pi \rho b^2\right)} \left[ \dot \theta \, b + \dot h \right] \sin \theta \end{array}, \end{equation}

while the $U_Y$ component is neglected since we have assumed previously that the presence of the virtual body prevents its motion in the $Y$ direction. It is worth highlighting that the normal component of the velocity in the body frame gives the essential contribution to obtain locomotion in the inertial frame. With respect to energy considerations, we may notice that the kinetic energy injected into the fluid by the body’s deformation is essentially transferred to the locomotion kinetic energy (see Wu Reference Wu1971, Kanso Reference Kanso2009). Since no vorticity release is included in the model, the total kinetic energy shows a periodically oscillating behaviour with a constant mean value, hence no power is transferred to the fluid and wasted into the wake. Consistently, the Froude efficiency, given in the present case by the ratio of two equal and null quantities, looses most of its interest and is not considered in the following. For a detailed description of the energy transfer process see Appendix C.

The motion of the body’s centre of mass is obtained by solving $ {{{\rm d} X}/{ {\rm d}t}} = U_X$ and integrating in time to obtain the axial displacement for each time.

From Appendix A we may notice that, even if $P_x = 0$ , this results in $F_x \neq 0$ , so the force is not purely normal to the body but a tangential component arises due to the rotation of the normal vector. This aspect has been mentioned in Limacher (Reference Limacher2021) discussing the chord-wise component of the added-mass force.

By recalling (A9) and using $\dot \theta$ for $\varOmega$ , we obtain

(2.18) \begin{equation} \left \{ \begin{array}{l} F_x = \pi \rho b^2\,\dot \theta \left[ u_y + \dot h + b \dot \theta \right],\\[2mm] F_y = - \pi \rho b^2\,\left[ \dot {u}_y + \ddot h + b \ddot \theta \right] ,\\[2mm] \end{array} \right . \end{equation}

where the expression of $F_y$ may be compared with the lift reported in Theodorsen (Reference Theodorsen1935). Let us underline that, by the theory of Theodorsen, we are able to compute separately the finite integral contributions given by the purely potential flow in terms of added-mass coefficients. By using the transpose of the rotation matrix (2.10) the forces in the ground and body reference frames are related by

(2.19) \begin{equation} \begin{bmatrix} F_X \\ F_Y \end{bmatrix} = \begin{bmatrix} \,\,\, \cos \theta \,F_x - \sin \theta \,F_y \\ \sin \theta \,F_x + \cos \theta \,F_y \end{bmatrix} \simeq \begin{bmatrix} \,\, \,F_x - \theta \,F_y \\ \theta \,F_x + \,F_y \end{bmatrix}. \end{equation}

The development of the forces during one period, both in the ground and in the body-fixed frame, is displayed in figure 2 for $h_o = 0.1$ and $\theta _o = 10^\circ$ . By evaluating the average values over one period we obtain $\overline {F_X} =0$ , $\overline {F_Y} =0$ , $\overline {F_y} =0$ while $\overline {F_x} \neq 0$ (represented by the dashed line in the same figure).

It should be noted that the forces described above arise solely from the flapping tail, with the added-mass contribution of the virtual body excluded from the balance.

Figure 2. Time behaviour of (a) $F_X$ and $F_Y$ in the ground frame; (b) $F_x$ and $F_y$ in the body-fixed frame (the dashed line shows the mean value of $F_x$ ) for $h_o=0.1$ and $\theta _o = 10^\circ$ .

3. Locomotion for prescribed heave and pitch

As a starting point, we evaluate the motion of the body which results from prescribed heave and pitch motions. The relevant parameters appearing in what follows are: the plate thickness $s^\ast$ , its length $c^\ast =2\,b^\ast$ and the densities of the fluid $\rho ^\ast$ and body $\rho ^\ast _b$ . While dimensional quantities were used up to this point, we will now distinguish between dimensional values, denoted with an asterisk ( $^\ast$ ), and their corresponding non-dimensional counterparts, which will be written without the asterisk. We define the amplitudes of the pitch and heave motions as $\theta _o$ and $h_o^\ast$ , respectively, while the phase angle between them is denoted as $\phi$ and the oscillation frequency is $\omega ^\ast = 2 \pi f^\ast$ . The motion of the plate is then given by

(3.1) \begin{equation} \left \{ \begin{array}{l} \theta = \theta _o \,\sin \left(\omega ^\ast \,t^\ast + \phi \right) ,\\[2mm] h^\ast = h_o^\ast \,\sin \left(\omega ^\ast \,t^\ast \right). \end{array} \right . \end{equation}

To determine the mean locomotion speed, $\overline {U^\ast _X}$ , we evaluate the average velocity over one period using (2.17). Noting that all cross-terms between heave and pitch contributions have a zero mean value, the only remaining term is $\overline {\theta \dot h^\ast } = (1/2) \omega ^\ast \theta _o h_o^\ast sin(\phi )$ , leading to

(3.2) \begin{equation} \overline {U^\ast _X} = \frac {1}{2} \frac {\pi \rho ^\ast {b^\ast }^2}{\left(m_b^\ast + \pi \rho ^\ast {b^\ast }^2\right)} \theta _o \omega ^\ast h_o^\ast \sin (\phi ). \end{equation}

From (3.2) we observe two key results: (i) the sign and magnitude of the mean locomotion velocity are directly controlled by $\sin \phi$ , the maximum locomotion (i.e. negative velocity) occurring for $\phi = -\pi /2$ , which means that the pitch motion lags the heave motion by a quarter period, in agreement with previous findings (Quinn, Lauder & Smits Reference Quinn, Lauder and Smits2015; Van Buren, Floryan & Smits Reference Van Buren, Floryan and Smits2018); (ii) no motion occurs when heave and pitch are in phase. Additionally, both heave and pitch must be present for locomotion. If either is absent, the body remains stationary.

By using $\omega ^\ast = 2\,\pi \,f^\ast$ and $h_o^\ast = b^\ast \, h_o$ , the average locomotion speed (3.2) can be expressed in terms of the reduced speed or the ‘stride length’, i.e. the non-dimensional locomotion velocity (based on the reference velocity $V^\ast _R = b^\ast f^\ast$ ) or the travelling distance measured in body lengths per second

(3.3) \begin{equation} U = \frac {\overline {U^\ast _X}}{b^\ast f^\ast } = \frac {\pi ^2 \rho ^\ast {b^\ast }^2}{\left(m_b^\ast + \pi \rho ^\ast {b^\ast }^2\right)} \theta _o h_o \sin (\phi ). \end{equation}

Figure 3. Time evolution of the $X$ -component of impulse, velocity and displacement for heave and pitch ( $h_o = 0.1$ ; $\theta = 10^\circ$ ; $\phi = -\pi /2$ ). The horizontal dashed line represents the mean value of $U_X$ .

Figure 3 illustrates the time evolution of impulse, velocity and displacement in the $X$ -direction for $h_o = 0.1$ and $\theta _0 = 10^\circ$ with a phase shift $\phi = -\pi /2$ . The dashed line in the figure represents the mean velocity calculated from (3.2). According to the analytical expression of the mean locomotion velocity (3.2), leftward motion occurs for negative phase angles. Figure 4 shows the solution in terms of $X$ -displacement in the ground frame as the phase shift between heave and pitch is varied from $0$ to $-\pi$ . In particular, we notice that the maximum displacement at the end of the cycle increases monotonically as $\phi$ approaches $-\pi /2$ , with smaller and equal values for symmetric variations of $\phi$ about $- \pi /2$ .

Figure 4. Temporal evolution of $X$ displacement for heave ( $h_o =0.1$ ) and pitch ( $\theta = 10^\circ$ ) with several values of the phase shift.

In previous studies (Paniccia et al. Reference Paniccia, Padovani, Graziani and Piva2021c , Reference Paniccia, Padovani, Graziani and Piva2023), it was shown that an appropriate phase shift can generate a travelling wave with an approximate phase velocity, $c=\omega ^\ast h_o^\ast /\theta _o$ , which reproduces a behaviour similar to undulatory swimming by creating a certain dissymmetry in the resulting motion. However, in the present case of a purely reactive solution, the resulting locomotion is significantly lower than the above defined $c$ , showing a poor performance in the absence of vortex shedding.

4. Locomotion of a passive flapping plate

In this section, we analyse the fluid dynamic moment acting on a flapping plate and derive the torque balance equation, which governs the evolution of passive pitch. To represent in an efficient way a flexible plate, we will assume that the pitching motion of the plate is not prescribed but results from a torque balance involving a torsional spring at the LE (see also Toomey & Eldredge Reference Toomey and Eldredge2008; Vanella et al. Reference Vanella, Fitzgerald, Preidikman, Balaras and Balachandran2009; Eldredge et al. Reference Eldredge, Toomey and Medina2010; Spagnolie et al. Reference Spagnolie, Moret, Shelley and Zhang2010; Zhang, Liu & Lu Reference Zhang, Liu and Lu2010; Moore Reference Moore2014; Hang et al. Reference Hang, Heydari, Costello and Kanso2022). To solve the torque balance, we must evaluate the hydrodynamic moment exerted by the surrounding fluid on the rigid body, which moves in an arbitrary manner in an unbounded fluid. For purely potential flow, only the reactive contribution is considered in the expression of the moment, while the circulatory contribution is omitted.

4.1. Fluid dynamic moment, torque balance and pitch solution

We start from the classical added-mass formulation as reported in Newman (Reference Newman2017), with an opposite sign convention due to the outward normal vector from the body into the fluid. We consider a two-dimensional flow field and use the position vector $r^\prime$ from the rotation centre at the LE of the plate. All the relevant details are reported in Appendix A.

The motion of the flat plate can be described in the body system as the superposition of the locomotion velocity components ( $u_x$ , $u_y$ ) and the prescribed heaving motion $\dot h$

(4.1) \begin{equation} U_1 = u_x + \dot h \sin \theta \simeq u_x + \dot h \theta \quad U_2 = u_y + \dot h \cos \theta \simeq u_y + \dot h. \end{equation}

By combining with (A17), removing quadratic terms and considering that $u_x = 0$ for a flat plate, we obtain

(4.2) \begin{equation} M_3 = - \pi \rho b^3 \!\left [\dot u_y + \ddot h + \frac {9}{8} b {\ddot \theta } \right ] ,\end{equation}

where the angular acceleration causes a moment which may be thought of as due to a virtual moment of inertia $ (9/8) \rho \pi b^4$ . By time differentiating (2.15) and combining with (4.2) we obtain

(4.3) \begin{equation} M_3 = -\pi \rho b^3 \!\left \{ \frac {m_b}{m_b + \pi \rho b^2} \ddot h + \left [ \frac {m_b}{m_b + \pi \rho b^2} + \frac {1}{8}\right ] b {\ddot \theta } \right \}. \end{equation}

Now we consider a thin foil undergoing prescribed heaving motion with a torsional spring at its LE. This set-up has been widely used in the literature as an effective model for plate flexibility due to fluid–body interaction, (see e.g. Moore Reference Moore2014). Since pitching motion is not prescribed, it must be determined by solving a torsional balance equation that accounts for the fluid dynamic moment, spring restoring torque and body inertia effects.

The goal is to analyse the combined effect of forced heaving and passive pitching motion in a perfect fluid to determine to what extent self-propulsion occurs. Although approximate, this model provides some insight into the body’s locomotion mechanism. A more generalised approach including the wake dynamics is addressed in an ongoing work.

A time-harmonic heaving motion, with non-dimensional amplitude $h_o ={h_o^\ast }/{b^\ast }$ , is imposed at the LE of the plate. In response to this driving motion, a torsional spring with rotational stiffness $k^\ast$ at the LE allows the plate to pitch passively with an assumed harmonic motion $\theta = {\theta _o} e^{i\,(\omega ^\ast \,t^\ast + \phi )}$ . The pitching motion of the plate (amplitude and phase) is not prescribed but rather is evaluated by balancing the angular moments due to the fluid (see 4.3), to the spring and to the body’s inertia.

The torque balance equation governing the passive flapping motion is derived in detail in Appendix B. Below, we only report its dimensionless form and the solution in terms of pitch amplitude and phase shift between heave and pitch

(4.4) \begin{equation} \frac {16}{3} R\,\ddot \theta + C \dot \theta + K\, \theta = {M_f} + {M_i} ,\end{equation}

where $R$ is the body–fluid mass ratio, $C$ and $K$ are the dimensionless spring damping and stiffness coefficients, respectively, $M_f$ is the dimensionless form of the fluid dynamic moment given by $M_3$ in (4.3) while $M_i$ is the inertial torque due to the forced heaving.

In the torsional balance (4.4), the terms related to $\dot \theta$ may introduce a non-zero phase shift between heave and pitch which is required to obtain $\theta \ne 0$ and hence locomotion. The $\dot \theta$ term clearly appears in the spring damping while, concerning the fluid dynamic moment $M_f$ , its presence would be associated with the circulatory component and vortex shedding.

Consistently, in the present purely reactive solution, this term is missing in the fluid dynamic moment, so the spring damping remains the only contributor to phase shift (see the second expression in (4.5) below).

Once the harmonic expressions for heave and pitch are considered and by using standard harmonic analysis, the solution for the passive pitch motion is obtained in terms of $\theta _o$ and $\phi$

(4.5) \begin{equation} \left \{ \begin{split} &\theta _o = \frac {\textit{BB}\,h_o}{\sqrt {\textit{AA}^2 + 4 \pi ^2 C^2}}, \\ &\tan(\phi ) = \frac {- 2 \pi C }{\textit{AA}} .\end{split} \right . \end{equation}

This solution is expressed in terms of the coefficients provided in Appendix B: $\textit{BB}$ , which depends on the mass ratio $R$ , and $AA$ , which is function of the mass ratio as well as of the spring stiffness $K$ . For the reader’s convenience, we report below their expressions

(4.6) \begin{align} \textit{BB} & = 16\,\pi ^2\,R \left [\frac {4\,R + 2\,\pi }{4\,R + \pi } \right ] \quad \quad R = \frac {s^\ast \,\rho ^\ast _b}{c^\ast \,\rho ^\ast },\end{align}

(4.7) \begin{align} \textit{AA} & = -4 \pi ^2 \left [ \frac {16}{3} R + \frac {\pi }{4} \left ( \frac {36\,R + \pi }{8\,R + 2\,\pi } \right ) \right ] + K = -4 \pi ^2 A_1 + K.\end{align}

Figure 5. Behaviour of the mean self-propulsion speed $U_s$ as a function of $K$ ( $h_o =0.1$ ) for different values of the spring damping coefficient $C$ : $C=0.1, 0.2, 0.5$ (a); $C=1, 2, 10.$ (b).

As clearly appears from (4.5), no pitch may occur without heave, i.e. if we have $h_o=0$ it follows also that $\theta _o=0$ . Moreover, the solution gives $\phi =\pm \pi /2$ for $AA=0$ , i.e., for a given relation between the mass ratio and the spring stiffness, which, as shown below, is related to the resonance frequency. In this condition the larger energy transfer is obtained, maximising locomotion, which still remains quite small.

4.2. Swimming speed and natural frequency

Once the pitch angle and phase have been evaluated through (4.5), then the mean locomotion speed $U_s = \overline {U_X}$ may be evaluated. By using the above introduced non-dimensional quantities, the first in (3.2) gives

(4.8) \begin{equation} U_s = \frac {\overline {U_X^\ast }}{b^\ast f^\ast } = \pi (1-\textit{CC}) \theta _o h_o \sin (\phi ) =\frac {\pi ^2}{4\,R + \pi }\, \theta _o h_o \sin (\phi ), \end{equation}

where the coefficient $\textit{CC}$ , depending on the mass ratio only, is reported in (B7) in Appendix B.

The above reported solution shows that: (i) no self-propulsion occurs unless pitch and heave are simultaneously present; (ii) a non-zero phase shift is required for the generation of net locomotion; and (iii) to achieve maximum velocity, $\phi \simeq \pm \pi /2$ is required. In the figures we present results for $U_s$ , $\theta _o$ and $\phi$ as functions of the spring stiffness $K$ for several values of the spring damping $C$ from 0.1 to 10. All the plots are drawn for a given value of $C$ , as a function of ${k^\ast }/{{f^\ast }^2}$ or $K$ in dimensionless form, to ensure that all solutions collapse onto a single curve (see also Zhong et al. Reference Zhong, Zhu, Fish, Kerr, Downs, Bart-Smith and Quinn2021). From figure 5 we may notice several results: first, the locomotion speed peaks at a specific value of $K$ and, second, the maximum value of the locomotion velocity increases as the spring damping decreases. This peak is likely a result of resonance effects, as discussed below.

Figure 6. Pitch amplitude $\theta _o$ as a function of $K$ ( $h_o =0.1$ ) for $C=0.1, 0.2, 0.5$ (a); $C=1, 2, 10.$ (b).

Figure 7. Phase angle variation as a function of $K$ ( $h_o =0.1$ ): (a) $C=0.1, 0.2, 0.5$ ; (b) $C=1, 2, 10$ .

The values of the pitch amplitude $\theta _o$ are reported in figure 6 as a function of $K$ for several values of $C$ . We notice both a peak at the same value of $K$ as for the locomotion speed and a reduction in the amplitude for increasing $C$ . Moreover, for $C$ larger than one, the pitch amplitude quickly falls to very small values. On the contrary, for the smallest $C$ it follows that $\theta _o$ exceeds the small-angle approximation, although a similar trend persists.

A comparison between active and passive motion reveals key differences. In the former case, the solution is given by (3.3) and shows a linear dependence of the velocity on both the amplitudes of heave $h_o$ and pitch $\theta _o$ . On the other hand, in the passive case the pitch amplitude is not prescribed but is evaluated from the torsional balance. Equation (4.5) shows that $\theta _o$ is linearly dependent on $h_o$ , leading to $U_s \propto h_o^2$ . Both the physical coefficients of the spring balance, $C$ and $K$ , can also significantly affect the resulting pitch amplitude. Thus, we may argue that, to achieve optimal locomotion, these physical parameters should be tuned consistently to achieve the proper pitch amplitude.

The variation of the phase angle $\phi$ between pitch and heave is plotted in figure 7 as a function of $K$ for the previously considered values of $C$ . Notably, a steep change in $\phi$ occurs near the peak values of both $U_s$ and $\theta _o$ . Specifically, by observing the peak values in figures 5 and 6 and by comparing with figure 7, we notice that, when $U_s$ and $\theta _o$ reach their maximum values, it follows always that $\phi = -\pi /2$ . In fact, from the second of (4.5), we notice that $\phi = \pm \pi /2$ for $AA=0$ or, due to (4.7), $K = 4\,\pi ^2\,A_1$ . By using the non-dimensional spring stiffness defined in (B4), the peak frequency occurs for

(4.9) \begin{equation} \frac {k^\ast }{{f^\ast }^2} = K \,\rho ^\ast \,{b^\ast }^4 = \rho ^\ast \,{b^\ast }^4\,4\,\pi ^2\,A_1 ,\end{equation}

where $A_1$ depends on the mass ratio only, and its expression is reported in Appendix B.

Figure 8. (a) Variation of $\theta _o$ and $\phi$ as a function of frequency; (b) self-propulsion velocity ( $h_o =0.1, k=4000, C=1$ ).

At the peak condition, i.e. for $AA=0$ , the torque balance equation shrinks to (see (B12))

(4.10) \begin{align} C\, \dot \theta = \textit{BB}\, h ,\end{align}

while the first expression in (4.5) yields

(4.11) \begin{align} \theta _o = \frac{\textit{BB}}{ 2 \pi C}\,h_o. \end{align}

Since the coefficient ${\textit{BB}}$ , defined in (4.6), is only function of the ratio between the body and the fluid mass, the pitch amplitude value increases without limits for vanishing spring damping $C$ . Then, from (4.8), the self-propulsion speed is expressed as

(4.12) \begin{align} U_s =- 16\,\pi ^3 \frac {R\,(2\,R + \pi )}{C\,(4\,R + \pi )^2} h_o^2 ,\end{align}

which also shows both the increase of the peak locomotion velocity for decreasing $C$ and the relevance of $h_o$ .

The pitch amplitude and phase and the opposite of the mean self-propulsion speed, $-U=-\overline {U_X^\ast }$ , for $C=1$ and $k=4000$ , are shown, as functions of the oscillation frequency, in figures 8(a) and 8(b), respectively. The values of $\theta _o$ and $\phi$ for some selected frequencies are used in figure 9 to represent the corresponding stroke loop in the shape space (i.e. where the variables $h$ and $\theta$ , which describe the shape of the body, are the independent variables). We notice that the area of each curve increases with the mean self-propulsion speed, while its orientation is only related to the phase angle. For vanishing $\phi$ , a reciprocal motion is obtained, which gives null locomotion. On the contrary, the curve with the largest area in the shape space corresponds to the maximum locomotion speed and to a phase shift $-\pi /2$ . These results confirm the observations of other authors (see e.g. Ross Reference Ross2006, Hatton & Choset Reference Hatton and Choset2010).

Figure 9. Stroke loops in the shape space for the frequencies selected in figure 8 ( $h_o =0.1, k=4000, C=1$ ).

Several solutions reported above for the passive swimming plate display a peaky behaviour which resembles that of the amplification factor of a forced harmonic oscillator with damping.

The peak frequency given by (4.9) is related to the natural oscillation frequency defined by (B19). From this expression we obtain $ {k^\ast }/{{f^\ast _N}^2} = 4 \pi ^2 \rho ^\ast {b^\ast }^4 A_1$ , as given by (4.9), confirming that the peak locomotion speed occurs at the natural frequency with a clear connection to resonance phenomena (see Beal et al. Reference Beal, Hover, Triantafyllou, Liao and Lauder2006, Michelin & Smith Reference Michelin and Smith2009, Alben et al. Reference Alben, Witt, Baker, Anderson and Lauder2012, Moore Reference Moore2014, Paraz, Schouveiler & Eloy Reference Paraz, Schouveiler and Eloy2016, Lopez-Tello, Fernandez-Feria & Sanmiguel-Rojas Reference Lopez-Tello, Fernandez-Feria and Sanmiguel-Rojas2023). The damping term was not accounted for in the above evaluation of the natural frequency, since it has a negligible effect.