3.1.1. Existence of an equilibrium lateral gap distance

In a side-by-side configuration, it has been known that for continuous swimming, the lateral interference of a nearby fin can enhance the schooling performance and a specific lateral spacing between two side-by-side fins exists to maintain a stable state as a result of flow-mediated interactions between the fins (Lin et al. Reference Lin, Wu, Zhang and Yang2021). However, the existence of the lateral equilibrium position for two side-by-side fins during intermittent swimming is still unclear. The cycle-averaged lateral force $(\overline {{F_y}} )$ acting on the upper fin for continuous swimming (CT mode) with f_a = 1.0 and intermittent swimming (HT, MTS and MTL modes) with DC = 0.5 is analysed in figure 3 as a function of the initial lateral gap distance (G ₀) to resolve the following questions: (i) Does an equilibrium lateral gap distance exist between two self-propelled side-by-side flexible fins with unique patterns in intermittent swimming and (if so) (ii) can two fins using a specific type of intermittent swimming reduce the equilibrium lateral gap distance compared with continuous swimming? In this figure, only data for the upper fin are presented because the overall lateral motions of the upper and lower fins are symmetric with respect to the centreline, resulting in identical cycle-averaged lateral forces $(\overline {{F_y}} )$ acting on the lower fin with an opposite sign (the overbar denotes the cycle-average value). In addition, the value of γ is set to 1.0. In figure 3, positive values of $\overline {{F_y}}$ acting on the upper fin in the CT mode (black circle) and the HT mode (red triangle) are found at small values of G ₀, indicating that the fins tend to push each other (away from the centreline). As the value of G ₀ increases, the values of $\overline {{F_y}}$ acting on the upper fin in the CT and HT modes become negative, and zero values of $\overline {{F_y}}$ are found at G ₀ = 2.4 for the CT mode and 2.8 for the HT mode, corresponding to the initial lateral gap distance at which an equilibrium stable state is achieved (i.e. G _0,eq). A similar trend of $\overline {{F_y}}$ is found on the upper fin for the MTL mode (green diamond). However, the value of G _0,eq for the MTL mode (G _0,eq = 1.82) is smaller than that for the CT mode. On the other hand, the values of $\overline {{F_y}}$ acting on the upper fin in the MTS mode (blue square) are always positive regardless of G ₀, indicating that two side-by-side flexible fins using this mode cannot achieve a stable propulsion state. At G ₀ = 8 for all modes, the values of $\overline {{F_y}}$ acting on the upper fin approach those of an isolated fin in each mode (dashed line) with little influence by the lower fin.

Figure 3.

Cycle-averaged lateral force $(\overline {{F_y}} )$ acting on the upper fin with γ = 1.0 as a function of the initial lateral gap distance (G ₀) for continuous swimming with f_a = 1.0 and intermittent swimming with DC = 0.5: black circle, CT mode; red triangle, HT mode; blue square, MTS mode; green diamond, MTL mode. Here, G _0,eq indicates the initial lateral gap distance at which an equilibrium stable state with $\overline {{F_y}} = 0$ is achieved in each mode. Blue and green dashed lines indicate the values of $\overline {{F_y}}$ acting on an isolated fin in the MT mode when the fin is coasting at the minimum Y_head and maximum Y_head, respectively. Black dashed line indicates a zero value of $\overline {{F_y}}$ acting on a single fin for the CT and HT modes.

In previous experimental and numerical studies, the existence of the lateral equilibrium positions for two side-by-side schooling fins with anti-phase oscillation and for a single fin near the ground has been widely reported when continuously prescribed heaving and/or pitching motions in the y-direction (constrained lateral motion) are adopted while a rigid fin is fixed in the x-direction (i.e. tethered systems) (Quinn et al. Reference Quinn, Moored, Dewey and Smits2014; Zhong et al. Reference Zhong, Han, Moored and Quinn2021) and while flexible fin(s) are free to move in the x-direction (i.e. self-propulsion systems) (Dai, He & Zhang Reference Dai, He and Zhang2016; Zhang, Huang & Lu Reference Zhang, Huang and Lu2017; Dai et al. Reference Dai, He, Zhang and Zhang2018b; Jeong et al. Reference Jeong, Lee and Park2021). It should be noted that flow and pressure fields around the upper fin with anti-phase oscillation in a side-by-side configuration are analogous to those around a single fin near the ground (Dewey et al. Reference Dewey, Quinn, Boschitsch and Smits2014; Zhong et al. Reference Zhong, Han, Moored and Quinn2021). Furthermore, the presence of equilibrium states for two side-by-side anti-phase flapping rigid fins and a single rigid fin near the ground during continuous swimming has been proved in theoretical studies with mathematical expressions for the hydrodynamic forces acting on rigid fin(s) with constrained lateral motions in tethered (Baddoo et al. Reference Baddoo, Kurt, Ayton and Moored2020, Reference Baddoo, Moore, Oza and Crowdy2021) and self-propulsion systems (Oza, Ristroph & Shelley Reference Oza, Ristroph and Shelley2019). Although the models using the constrained lateral motions above are unable to move toward an equilibrium lateral position while reacting to the surrounding fluid, an idealized pure pitching motion of two side-by-side anti-phase flapping rigid fins and a single rigid fin under a ground effect to move freely in the y-direction has shown flow-mediated organization (at equilibrium positions) in the y-direction (Kurt et al. Reference Kurt, Cochran-Carney, Zhong, Mivehchi, Quinn and Moored2019) and in the x- and y-directions (Kurt et al. Reference Kurt, Ormonde, Mivehchi and Moored2021; Lin et al. Reference Lin, Wu, Zhang and Yang2021). However, it should be noted that the propulsive performance in this case is underestimated due to the dynamic recoil motion (i.e. opposite lateral motion of the leading and trailing edges) that occurs (Kurt et al. Reference Kurt, Cochran-Carney, Zhong, Mivehchi, Quinn and Moored2019, Reference Kurt, Ormonde, Mivehchi and Moored2021; Lin et al. Reference Lin, Wu, Zhang and Yang2021). Based on these observations, it is reasonable to infer that the occurrence of equilibrium lateral states is a common phenomenon for two side-by-side anti-phase flapping fins and a single fin near the ground during continuous swimming regardless of the conditions of the model set-up, such as flexibility, moving or stationary conditions and types of active motions. In contrast to continuous swimming, however, although three types of intermittent swimming modes for two fins (MTL, MTS and HT modes) are considered, only the MTL and HT modes can present an equilibrium state, demonstrating that two side-by-side flexible fins in a proper form of intermittent swimming motion can maintain an equilibrium state.

Because the cycle-averaged lateral gap distance between two fins in equilibrium states is affected by the intermittent swimming mode (for example, see the MTS and MTL modes in figure 2), an equilibrium lateral gap distance (G_eq) is derived mathematically by averaging the temporal lateral gap distance between the upper and lower fins (G) during the cycle for an equivalent comparison of the lateral equilibrium distances between different modes: G_eq = G _0,eq − A_head for the CT and HT modes, G_eq = G _0,eq − A_headDC for the MTL mode and G_eq = G _0,eq − A_headDC − 2A_head for the MTS mode. The values of G_eq for the CT, HT and MTL modes are 2.0, 2.4 and 1.62, respectively.

At the equilibrium positions (G_eq), the time histories of the lateral position of the leading edge (Y_head) and trailing edge (Y_tail) are presented in figure 4 to show the flapping kinematics of the upper fin in each mode. In figure 4(a), the profile of Y_tail (blue line) for the upper fin in the CT mode is symmetric with respect to the equilibrium position of the prescribed lateral motion of Y_head (here, G_eq/2; see the horizontal black dashed line). It should be noted that the profile of Y_tail of two side-by-side flexible fins (or a single flexible fin near the ground) during continuous swimming can be asymmetric when the lateral gap distance between the fins (or from the ground) is reduced (G ₀ < G_eq) due to the strong influence of a nearby fin (or ground) (Dai et al. Reference Dai, He and Zhang2016; Park et al. Reference Park, Kim and Sung2017). Similar to the CT mode, a symmetric profile of Y_tail for the upper fin in the HT mode with respect to G_eq/2 can be found in figure 4(b). These results indicate that two side-by-side flexible fins in the CT and HT modes maintain the symmetric flapping kinematics when they achieve a stable lateral state despite the passive response of the fins. In contrast to the observations for the CT and HT modes, two side-by-side flexible fins in the MTL mode, as depicted in figure 4(c), show asymmetric flapping kinematics for Y_tail with respect to G_eq/2. In addition, the flapping kinematics for the MTL mode is asymmetric even with respect to the equilibrium positions of bursting (green dashed line during 0 ≤ t/T_cyc < 0.5) and coasting (green dashed line during 0.5 ≤ t/T_cyc < 1.0) motions. The inconsistent flapping kinematics between the HT and MTL modes suggests that the symmetric nature of the body response for two side-by-side flexible fins during intermittent swimming is strongly influenced by the coasting position and arrangement (figure 4b,c). In the following sections, a two-fin system in the MTL mode is studied further as a representative of two side-by-side flexible fins during intermittent swimming because it is expected that the smaller value of G_eq for the MTL mode than that for the HT mode (figure 3) leads to a performance enhancement of two self-propelled side-by-side flexible fins with anti-phase oscillation (ϕ = ${\rm \pi}$) (Dai et al. Reference Dai, He and Zhang2016; Park et al. Reference Park, Kim and Sung2017; Zhang et al. Reference Zhang, Huang and Lu2017; Peng et al. Reference Peng, Huang and Lu2018).

Figure 4.

Time histories of the lateral position of the leading edge (Y_head) and trailing edge (Y_tail) for the upper fin at G_eq when γ = 1.0: (a) CT mode with f_a = 1.0, (b) HT mode with DC = 0.5 and (c) MTL mode with DC = 0.5. Horizontal black dashed lines indicate half values of G_eq in each mode. Two numbers in each mode indicate the maximum and minimum values of Y_tail. In (b,c), white and grey regions indicate the burst and coast phases. Green dashed lines during the burst period (0 ≤ t/T_cyc < 0.5) and the coast period (0.5 ≤ t/T_cyc < 1.0) in (c) denote the equilibrium positions of each period for the MTL mode.

3.1.2. Hydrodynamic mechanism leading to lateral stable equilibrium states

In this section, we investigate a hydrodynamic mechanism leading to lateral equilibrium states for two side-by-side flexible fins during the intermittent swimming with a focus on the following two questions: (i) What components play a role in the achievement of a lateral stable state in the MTL mode, and how does the hydrodynamic mechanism leading to lateral equilibrium states for the MTL mode differ from that for the CT mode, and (ii) why is a stable equilibrium state for two flexible fins in the MTL mode achieved at a lateral gap distance smaller than that in the CT mode? To identify the components of the hydrodynamic forces acting on a fin, a control volume (CV) analysis in the y-direction is employed since the force production of flapping fins can be extracted from the velocity and pressure fields within a CV (Kurtulus, Scarano & David Reference Kurtulus, Scarano and David2007; Jardin, David & Farcy Reference Jardin, David and Farcy2009; Shen, Chan & Lin Reference Shen, Chan and Lin2009; Wu et al. Reference Wu, Shu, Wan, Wang and Chen2022). The detailed process of the CV analysis is presented below. The conservation of the momentum in a moving CV is expressed as

(3.1)\begin{align}&\frac{\partial }{{\partial {t^\ast }}}\int \rho _0^\ast \boldsymbol{u}^{\ast}\,\textrm{d}{\forall}^\ast + \oint {({\boldsymbol{n}_{out}} \cdot \rho _0^\ast {\boldsymbol{u}^\ast })({\boldsymbol{u}^\ast } - \boldsymbol{u}_{cv}^\ast )\,\textrm{d}{S^\ast }}\nonumber\\ &\quad ={-} \oint {{p^\ast }{\boldsymbol{n}_{out}}\,\textrm{d}{S^\ast }} + \oint {{\tau ^\ast }{\boldsymbol{n}_{out}}\,\textrm{d}{S^\ast }} - \boldsymbol{F}_L^\ast ,\end{align}

where ${\forall} ^\ast $ is the CV, S* is the control surface (CS), n_out is the outward-facing unit normal vector on the surface, $\boldsymbol{u}_{cv}^\ast $ is the velocity of the CV and τ* is the viscous stress tensor. For two-dimensional flows, the momentum equation along the y-direction is written as

(3.2) \begin{align}& \dfrac{\partial }{{\partial {t^\ast }}}\int_{\forall} {\rho _0^\ast {v^\ast }\,\textrm{d}{\forall} ^\ast } + \int_{S1} {\rho _0^\ast {u^\ast }({v^\ast } - v_{cv}^\ast )\,\textrm{d}{S^\ast }} + \int_{S2} {\rho _0^\ast {v^\ast }({v^\ast } - v_{cv}^\ast )\,\textrm{d}{S^\ast }}\nonumber\\ &\quad\quad - \int_{S3} {\rho _0^\ast {u^\ast }({v^\ast } - v_{cv}^\ast )\,\textrm{d}{S^\ast }} - \int_{S4} {\rho _0^\ast {v^\ast }({v^\ast } - v_{cv}^\ast )\,\textrm{d}{S^\ast }}\nonumber\\ & \quad = \int_{S1} {{\mu ^\ast }\left( {\dfrac{{\partial {v^\ast }}}{{\partial {x^\ast }}} + \dfrac{{\partial {u^\ast }}}{{\partial {y^\ast }}}} \right)\,\textrm{d}{S^\ast }} + \int_{S2} {\left( { - {p^\ast } + 2{\mu^\ast }\dfrac{{\partial {v^\ast }}}{{\partial {y^\ast }}}} \right)\ \textrm{d}{S^\ast }} \nonumber\\ & \quad\quad - \int_{S3} {{\mu ^\ast }\left( {\dfrac{{\partial {v^\ast }}}{{\partial {x^\ast }}} + \dfrac{{\partial {u^\ast }}}{{\partial {y^\ast }}}} \right)\,\textrm{d}{S^\ast }} - \int_{S4} {\left( { - {p^\ast } + 2{\mu^\ast }\dfrac{{\partial {v^\ast }}}{{\partial {y^\ast }}}} \right)\,\textrm{d}{S^\ast }} - F_y^\ast , \end{align}

where S1~S4 indicate the CSs at the outlet, top, inlet and bottom boundaries, respectively (see a schematic of a CV around a fin in figure 5). The lower and upper limits of integration for the CSs S1 and S3 are $y_{S4}^\ast $ and $y_{S2}^\ast $ and those for the CSs S2 and S4 are $x_{S3}^\ast $ and $x_{S1}^\ast $. This equation is non-dimensionalized by the following characteristic scales: the fin length L* for the length, the reference velocity $U_{ref}^\ast $ for the velocity, ${L^\ast }/U_{ref}^\ast $ for the time t*, $\rho _0^\ast {(U_{ref}^\ast )^2}$ for the pressure p* and $\rho _0^\ast {L^\ast }{(U_{ref}^\ast )^2}$ for the force $F_y^\ast $. The dimensionless momentum equation can then be summarized as follows:

(3.3)\begin{align}{F_y} & = \underbrace{{\left( { - \dfrac{\partial }{{\partial t}}\int_{\forall} {v\,\textrm{d}{\forall} } } \right)}}_{{F_y^u:\textit{unsteady}\ \textit{term}}} + \underbrace{{\left( {\int_{S2} {( - p)\,\textrm{d}S} - \int_{S4} {( - p)\,\textrm{d}S} } \right)}}_{{F_y^p:\textit{pressure}\ \textit{term}}}\nonumber\\ & \quad + \underbrace{{\left( { - \int_{S1} {u\textrm{(}v - {v_{cv}}\textrm{)}\,\textrm{d}S} - \int_{S2} {v(v - {v_{cv}})\,\textrm{d}S} + \int_{S3} {u\textrm{(}v - {v_{cv}}\textrm{)}\,\textrm{d}S} + \int_{S4} {v(v - {v_{cv}})\,\textrm{d}S} } \right)}}_{{F_y^m:\textit{momenterm}\ \textit{flux}\ \textit{term}}}\nonumber\\ & \quad + \underbrace{{\left( {\dfrac{1}{{Re}}\int_{S1} {\left( {\dfrac{{\partial v}}{{\partial x}} + \dfrac{{\partial u}}{{\partial y}}} \right)\,\textrm{d}S} + \dfrac{2}{{Re}}\int_{S2} {\dfrac{{\partial v}}{{\partial y}}\,\textrm{d}S} - \dfrac{1}{{Re}}\int_{S3} {\left( {\dfrac{{\partial v}}{{\partial x}} + \dfrac{{\partial u}}{{\partial y}}} \right)\,\textrm{d}S} - \dfrac{2}{{Re}}\int_{S4} {\dfrac{{\partial v}}{{\partial y}}\,\textrm{d}S} } \right)}}_{{F_y^s:\textit{shear}\ \textit{stress}\ \textit{term}}}, \end{align}

where $F_y^u$, $F_y^p$, $F_y^m$ and $F_y^s$ are force components related to the unsteady, pressure, momentum flux and shear stress effects in a CV. Further details are available in the literature (Kurtulus et al. Reference Kurtulus, Scarano and David2007; Jardin et al. Reference Jardin, David and Farcy2009; Shen et al. Reference Shen, Chan and Lin2009). To investigate the effects of the velocity and pressure fields acting on two side-by-side fins in a self-propulsion system, a moving and non-deformable CV around the upper fin is determined close to the fin, similar to that adopted in a recent study of an unconstrained flapping fin by Wu et al. (Reference Wu, Shu, Wan, Wang and Chen2022): x_{S 1} = X_head,u + 3, y_{S 2} = (G ₀/2 − A_head/2) + 0.6, x_{S 3} = X_head,u − 0.5 and y_{S 4} = (G ₀/2 − A_head/2) − 0.6. In the reference frame, CV and CS translate with U_avg along the moving direction; i.e. u_cv = (u_cv, v_cv) = (−U_avg, 0) (figure 5), where U_avg is the average horizontal speed of a fin $({U_{avg}} = \int_0^1 {|\partial X/\partial t|\,\textrm{d}s} )$. Here, the lateral spacing of the CV is chosen to cover the flapping range of the upper fin in the CT and MTL modes.

Figure 5.

Schematic of the CV around a self-propelled flexible fin. The reference frame of the CV and CS translates with the horizontal speed of the fin (U_avg) toward the negative x-direction. Green arrows indicate induced flows passing through the CSs. During the cycle, the surrounding fluid bleeds into the CV through CSs S2 and S4 (see inward green arrows) and then leaves the CV through CS S1 for propulsion of the fin in the negative x-direction (see outward green arrows).

Figure 6 displays the time histories of estimated values for components consisting of the lateral force (F_y) acting on the upper fin at G_eq in the CT mode with f_a = 1.0 and in the MTL mode with DC = 0.5. As shown, the sum of the unsteady $(F_y^u)$, momentum flux $(F_y^m)$, pressure $(F_y^p)$ and shear stress $(F_y^s)$ terms in figure 6(a,b) is very similar to the value of F_y, indicating that the lateral force acting on the fin can be decomposed into these four terms. For the upper fin in the CT mode in figure 6(a), the profiles of $F_y^m$ and $F_y^p$ with the cycle-averaged values of $\overline {F_y^m}$ and $\overline {F_y^p}$ (see the inset) demonstrate the dominance of the negative value of $F_y^m$ and the positive value of $F_y^p$ during the cycle, with the balance between the two circulatory lateral force terms, i.e. the momentum flux and pressure terms ($\overline {F_y^m}$ and $\overline {F_y^p}$), leading to a zero cycle-averaged lateral force $(\overline {{F_y}} = 0)$. Similarly, for the MTL mode in figure 6(b), the two circulatory lateral force terms of $\overline {F_y^m}$ and $\overline {F_y^p}$ are important for achieving an equilibrium state. However, it is evident that a negative unsteady term $(\overline {F_y^u} )$ (i.e. added-mass lateral force) contributes to the formation of the lateral equilibrium state for the upper fin. Because this negative unsteady term is a key factor to reduce the equilibrium lateral gap distance between two fins in the MTL mode, the value of G_eq for the MTL mode is smaller than that for the CT mode; if the value of $\overline {F_y^u}$ is zero for the two fins in the MTL mode, a lateral equilibrium state occurs at G _0,eq = 2.8 (i.e. G_eq = 2.6), a value larger than that for the CT mode in figure 3. The contribution of the shear stress term to the generation of the lateral force acting on the upper fin in each mode is negligible. Below (figures 7–10), we present the detailed generation process of each component comprising the lateral force acting on two fins in the CT and MTL modes.

Figure 6.

Time histories of estimated values for components comprising the lateral force (F_y) acting on the upper fin at G_eq when γ = 1.0: red, $F_y^u$ (unsteady term); blue, $F_y^m$ (momentum flux term); green, $F_y^p$ (pressure term); pink, $F_y^s$ (shear stress term); black, F_y. For comparison, the sum of the four components in each mode is inserted with circle symbols: (a) CT mode with f_a = 1.0 and G_eq = 2.0 and (b) MTL mode with DC = 0.5 and G_eq = 1.62.

Figure 7.

Cycle-averaged lateral velocity contours around two fins at G_eq when γ = 1.0: (a) CT mode with f_a = 1.0 and (b) MTL mode with DC = 0.5. Green bold lines represent the fins.

Figure 8.

(a) Instantaneous vorticity contours around two fins at G_eq in the MTL mode with DC = 0.5 and γ = 1.0. Black bold lines represent the fins. (b) Temporal lateral positions of the leading edge (Y_head) (red) and trailing edge (Y_tail) (blue) for two fins. (c) Schematic of two consecutive vortices to explain the vortex-induced velocity ($\textbf{U}_{\varGamma}$) based on the vortex dipole model. Four instances (i–iv) in (a) are chosen based on the local maximum and minimum Y_tail in (b). The four vortices behind the upper fin in (a) are denoted by I, II, III and IV based on the sequence of vortex shedding (from I to IV), and the induced velocities between vortices I and II at instance (ii), between vortices II and III at instance (iii) and between vortices III and IV at instance (iv) are denoted by the blue arrows. In (a), the induced horizontal velocities between the upper and lower fins are denoted by the red arrows. The lengths of the blue and red arrows are proportional to the strengths of the induced velocities.

Figure 9.

(a,b) Instantaneous pressure contours around two fins at G_eq with γ = 1.0 when the lateral force (F_y) acting on the upper fin reaches its (i) maximum and (ii) minimum: (a) CT mode with f_a = 1.0 and (b) MTL mode with DC = 0.5. In (a,b), green bold lines represent the fins. (c,d) Temporal variations of the lateral force (F_y) (red) and the average lateral velocity (V_avg) (blue) for each case. (e,f) Conditionally averaged (e) positive pressure $(\overline {{p^{pos}}} )$ and (f) negative pressure $(\overline {{p^{neg}}} )$ along the lower surface of the upper fin at G_eq: red circle, CT mode with f_a = 1.0; blue triangle, MTL mode with DC = 0.5. For comparison, data for an isolated fin in the CT mode with f_a = 1.0 (red dashed line) and MT mode (coasting at the maximum Y_head) with DC = 0.5 (blue dashed line) are included.

Figure 10.

Time histories of (a,b) the lateral position of the leading edge (Y_head) and the average lateral position (Y_avg) and (c,d) the average lateral acceleration $({\dot{V}_{avg}})$ for the upper fin at G_eq when γ = 1.0: (a,c) CT mode with f_a = 1.0 and (b,d) MTL mode with DC = 0.5. White and grey regions in each figure indicate acceleration states toward upward $({\dot{V}_{avg}} > 0)$ and downward $({\dot{V}_{avg}} < 0)$ directions, respectively. Horizontal dashed line indicates a zero value.

To examine why the negative cycle-averaged momentum flux term $(\overline {F_y^m} )$ occurs for the two fins in the CT and MTL modes, the cycle-averaged lateral velocity contours around two fins at G_eq are presented in figure 7. The contours behind the upper fin in the CT and MTL modes show the dominance of the positive cycle-averaged lateral velocity (red contour). Because a positive horizontal velocity (u) behind a fin is expected to occur to propel toward the negative x-direction, an upward deflected jet with the dominance of a positive lateral velocity behind a fin during the cycle acts to induce a negative $\overline {F_y^m}$ (see the momentum flux term at the CS S1 in (3.3)). It should be noted that because the momentum flux term in the y-direction is coupled with the horizontal velocity (3.3), the jet strength shown in figure 7 is not directly proportional to the magnitude of $\overline {F_y^m}$.

Figure 8 shows the instantaneous vorticity contours around two fins at G_eq for the MTL mode with DC = 0.5 to explain the formation process of the cycle-averaged lateral velocity for intermittent swimming (figure 7b). Four instances (i‒iv) in figure 8(a) are selected based on the local maximum and minimum Y_tail (see figure 8b). In figure 8(a), four vortices (I‒IV) are generated by the upper fin over each cycle period, and they induce local lateral velocities (see the blue arrows), i.e. positive lateral velocities by vortex pairs (I, II) and (III, IV) and negative lateral velocity by a vortex pair (II, III). The horizontal flows between the counter-rotating vortices generated by the upper and lower fins (see the red arrows) lead to the downstream advection of vortices II and IV and the upstream advection of vortex III, resulting in an increase (decrease) of the positive (negative) lateral velocities. For a comparison between the vortex-induced positive and negative lateral velocities in terms of the magnitude, the vortex dipole model proposed by Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009) is employed. Based on the vortex dipole model (figure 8c), the vortex-induced horizontal ($U_{\varGamma}$) and lateral ($V_{\varGamma}$) velocities can be approximated as ${\boldsymbol{U}_\varGamma } = ({U_\varGamma },{V_\varGamma }) = ((\varGamma /2{\rm \pi} d)\cos {\alpha ^\ast },(\varGamma /2{\rm \pi} d)\sin {\alpha ^\ast })$, where ${\boldsymbol{U}_\varGamma } = \boldsymbol{U}_\varGamma ^\ast{/}U_{ref}^\ast $, $\varGamma = {\varGamma ^\ast }/(U_{ref}^\ast {L^\ast })$, $d = {d^\ast }/{L^\ast }$ and α* are correspondingly the induced velocity, the average circulation of two vortices in the dipole, the distance between the vortex cores, and the orientation angle of $\textbf{U}_{\varGamma}$ relative to the x-direction (Godoy-Diana et al. Reference Godoy-Diana, Marais, Aider and Wesfreid2009; Zheng & Wei Reference Zheng and Wei2012; Kim & Lee Reference Kim and Lee2019). Each value of ${\varGamma}$, d and α* is estimated when an upstream vortex in a dipole is shed from the trailing edge of the fin. The vortex circulation is calculated through the integration of the vorticity over a rectangular area, and the integration area is determined by performing Gaussian fits $({\textrm{e}^{( - x_i^2/\sigma _i^2)}})$ along the x- and y-directions based on the positions of the vorticity maximum and minimum. The size of each vortex along the x- and y-directions is defined as 2σ_i (Godoy-Diana et al. Reference Godoy-Diana, Marais, Aider and Wesfreid2009; Zheng & Wei Reference Zheng and Wei2012). For dipole (I, II) at instance (ii), dipole (II, III) at instance (iii) and dipole (III, IV) at instance (iv) in figure 8(a), the estimated values of ${\varGamma_{j}}$, d_j, $\alpha _j^\ast $ and $V_{\varGamma j}$, respectively, are listed in the inset, where the subscript j represents a value formed between the jth vortex and the j + 1th vortex (for example, d ₁ is the distance between vortices I and II). A comparison between the vortex-induced positive and negative lateral velocities (see the insets in figure 8a) demonstrates that the presence of the positive cycle-averaged lateral velocity behind the upper fin in the MTL mode in figure 7(b) is closely associated with the large contribution of the positive $V_{\varGamma}$₁ and $V_{\varGamma}$₃ compared with that of the negative $V_{\varGamma}$₂.

The overall process for the creation of vortices and their induction of local lateral velocities for the upper fin in the MTL mode is similar to previous findings pertaining to two side-by-side anti-phase flapping rigid/flexible fins (Dewey et al. Reference Dewey, Quinn, Boschitsch and Smits2014; Peng et al. Reference Peng, Huang and Lu2018) and for a single rigid/flexible fin near the ground (Quinn et al. Reference Quinn, Moored, Dewey and Smits2014; Dai et al. Reference Dai, He and Zhang2016; Park et al. Reference Park, Kim and Sung2017; Kurt et al. Reference Kurt, Cochran-Carney, Zhong, Mivehchi, Quinn and Moored2019) during continuous swimming. Furthermore, the role of the induced horizontal flows on the increase (decrease) of the positive (negative) lateral velocities between vortices for the upper fin in the MTL mode is similar to our observation from the CT mode (not shown here for brevity). Nonetheless, a careful inspection of the vortex patterns in figure 7 shows less coherence of the vortex organization behind the upper fin in the MTL mode than that in the CT mode due to the presence of a coasting period in the MTL mode. Furthermore, the strength of the lateral jet deflection is greater in the MTL mode than in the CT mode in figure 7 due to a large difference between the vortex-induced positive and negative lateral velocities right behind the upper fin during the cycle. Temporal evolutions of the vortex structures for the generation of an upward deflection jet behind the upper fin in the MTL mode in figure 8(a) are found for all DC (0.1 ≤ DC < 1.0), consistent with a previous experimental study of a single pitching fin under intermittent swimming (MT mode) (Floryan, Buren & Smits Reference Floryan, Buren and Smits2017). However, the properties of vortex structures (i.e. ${\varGamma}$, d and α*) and their advection speeds vary according to the intermittency (i.e. DC).

Next, in order to explain the generation of the positive cycle-averaged pressure term $(\overline {F_y^p} )$ for the upper fin in the CT and MTL modes in figure 6, the instantaneous pressure contours around two fins at G_eq are presented in figure 9(a,b). Figure 9(c,d) show the temporal variations of the lateral force (F_y) (red) and the average lateral velocity (V_avg) (blue) for the upper fin, where the average lateral velocity (V_avg) along a fin is defined as ${V_{avg}} = \int_0^1 {(\partial Y(s,t)/\partial t)\,\textrm{d}s}$. The pressure contours in each mode are extracted when the lateral force (F_y) acting on the upper fin reaches its maximum (figure 9a-i,b-i) and minimum (figure 9a-ii,b-ii) (see the vertical dashed lines in figure 9c,d). Consistent with previous findings for two side-by-side flexible fins during continuous swimming (Peng et al. Reference Peng, Huang and Lu2018), the positive (figure 9a-i,b-i) and negative pressures (figure 9a-ii,b-ii) around two fins are enhanced by sharing the pressure field for all modes during the inward and outward motions, respectively (figure 9c,d). The enhancement of the pressures around two side-by-side fins by schooling can be estimated through a comparison with those around a single fin in each mode (i.e. CT mode and MT mode coasting at the maximum Y_head; see only the red line in figure 2a,d). To quantify the enhancement of each component of the positive and negative pressure, conditional averaging of the positive and negative pressure during the cycle is performed; these results ($\overline {{p^{pos}}}$ and $\overline {{p^{neg}}}$) are plotted in figure 9(e,f). Here, $\overline {{p^{pos}}}$ and $\overline {{p^{neg}}}$ are the conditionally averaged positive and negative pressure along the under surface of the upper (single) fin, respectively, during the cycle. A comparison of the conditionally averaged pressure between the upper fin and single fin shows that the increasing magnitude of the positive pressure by schooling in each mode exceeds that of the negative component, leading to a positive $\overline {F_y^p}$ for the upper fin in the CT and MTL modes in figure 6. The dominance of the enhanced positive pressure for the upper fin in the CT and MTL modes in figure 9 is consistent with a previous observation, commonly known as the ‘cushion effect’, which serves to lift an actively flapping rigid/flexible body and a stationary body near the ground (Zhang et al. Reference Zhang, Huang and Lu2017; Kurt et al. Reference Kurt, Cochran-Carney, Zhong, Mivehchi, Quinn and Moored2019; Jeong et al. Reference Jeong, Lee and Park2021). On the other hand, the increase of $\overline {{p^{pos}}}$ for the upper fin in the MTL mode relative to a single fin in the MT mode (see the blue colouring in figure 9e) is larger than that in the CT mode due to the small value of G_eq, suggesting a large difference of the values of $\overline {F_y^p}$ between the single and upper fins in the MT/MTL mode compared with that in the CT mode ($\Delta \overline {F_y^p} = 0.158$ for the CT mode and 0.168 for the MT/MTL mode).

Finally, to analyse how the negative cycle-averaged unsteady term $(\overline {F_y^u} )$ in figure 6 arises for intermittent swimming (MTL mode), the time histories of the lateral position of the leading edge (Y_head), the average lateral position $({Y_{avg}} = \int_0^1 {Y(s,t)\,\textrm{d}s} )$ and the average lateral acceleration $({\dot{V}_{avg}} = \textrm{d}{V_{avg}}/\textrm{d}t)$ of the upper fin at G_eq during the cycle are presented in figure 10. Here, the lateral position of Y_head and Y_avg for the upper fin in the CT and MTL modes is modulated by subtracting G _0,eq/2 − A_head/2. For the upper fin in the CT mode, the average lateral position (Y_avg) in figure 10(a) shows a symmetric profile during the cycle (consistent with the variation of Y_tail in figure 4a), and thus the balanced variation of ${\dot{V}_{avg}}$ between positive and negative values during the cycle in figure 10(c) results in the zero $\overline {F_y^u}$ shown in figure 6. Contrary to the observation in the CT mode, the profiles of Y_avg and ${\dot{V}_{avg}}$ for the MTL mode are complicated with asymmetric motions during the cycle in figure 10(b,d) (also see figure 4c). The signs of ${\dot{V}_{avg}}$ are negative (grey regions) and positive (white regions) when the lateral positions of Y_avg for the upper fin are above and below the equilibrium positions (Y = 0 during the burst period (0 ≤ t/T_cyc < 0.5) and Y = 0.2 during the coast period (0.5 ≤ t/T_cyc < 1.0)). This relationship between Y_avg and ${\dot{V}_{avg}}$ can be explained by a restoring force acting in the direction of the equilibrium positions under the prescribed bursting or coasting motion. Interestingly, the magnitude of the positive ${\dot{V}_{avg}}$ is larger than that of the negative ${\dot{V}_{avg}}$ during the burst period because the large lateral displacement of Y_avg near t/T_cyc = 0.3 (from the equilibrium position) due to the elastic response by the passive flexibility generates a strong restoring force acting in the upward direction. This result indicates that the negative value of $\overline {F_y^u}$ for the upper fin in the MTL mode in figure 6 is attributable to the positive lateral fin (or flow) accelerations (see the unsteady term in (3.3)) caused by the strong restoring force. During the coast period, the values of ${\dot{V}_{avg}}$ are relatively small due to the weakened restoring force as the lateral spacing between Y_avg and the equilibrium position (Y = 0.2) is reduced. To support our observation in figure 10, the root-mean-square of the conditional lateral accelerations (${(\dot{V}_{avg}^{pos})_{rms}}$ and ${(\dot{V}_{avg}^{neg})_{rms}}$) during the cycle is calculated, where $\dot{V}_{avg}^{pos}$ and $\dot{V}_{avg}^{neg}$ are the positive and negative components of the lateral accelerations of a fin, respectively. The estimated value of ${(\dot{V}_{avg}^{pos})_{rms}}$ for the upper fin in the MTL mode (=0.637) is found to be larger than that of ${(\dot{V}_{avg}^{neg})_{rms}}$ (=0.564), indicating the dominance of the positive lateral fin acceleration for intermittent swimming.

The hydrodynamic mechanism leading to lateral equilibrium states of fin(s) for continuous swimming was examined in an experimental study of a single pitching rigid fin near the ground in a tethered system (Kurt et al. Reference Kurt, Cochran-Carney, Zhong, Mivehchi, Quinn and Moored2019). Kurt et al. (Reference Kurt, Cochran-Carney, Zhong, Mivehchi, Quinn and Moored2019) found that a stable equilibrium lateral position of a pitching rigid fin on the ground is mainly determined by the balance between the enhanced positive pressures and jet deflections; moreover, the contribution of the negative cycle-averaged added-mass lateral force on the equilibrium state is small. Despite the different model set-ups (e.g. self-propulsion, passive flexibility and heaving motion), the underlying mechanism leading to an equilibrium state for two side-by-side flexible fins in the CT mode (figures 7, 9 and 10) is similar to that for a single rigid fin near the ground as found by Kurt et al. (Reference Kurt, Cochran-Carney, Zhong, Mivehchi, Quinn and Moored2019). Together with previous observations using two side-by-side anti-phase flapping rigid/flexible fins in tethered (Dewey et al. Reference Dewey, Quinn, Boschitsch and Smits2014) and self-propulsion systems (Peng et al. Reference Peng, Huang and Lu2018; Lin et al. Reference Lin, Wu, Zhang and Yang2021) and a single rigid/flexible fin in fixed (Quinn et al. Reference Quinn, Moored, Dewey and Smits2014; Zhong et al. Reference Zhong, Han, Moored and Quinn2021) and free-swimming conditions (Dai et al. Reference Dai, He and Zhang2016; Park et al. Reference Park, Kim and Sung2017; Zhang et al. Reference Zhang, Huang and Lu2017) under continuous heaving or pitching motions, the implication of the present findings is that the circulatory lateral forces caused by the enhanced positive pressures and jet deflections play a dominant role in determining the lateral equilibrium state of fin(s) in continuous (symmetric) swimming regardless of the model set-up.

In contrast, for the MTL mode, the relationship between the lateral position Y_avg and the acceleration ${\dot{V}_{avg}}$ in figure 10(b,d) indicates that the asymmetric flapping kinematics of the average lateral motion during the cycle is important in the generation of the added-mass lateral force. However, it should be noted that for two side-by-side rigid fins in the MTL mode with no passive flexibility (Y_avg = Y_head), the lateral equilibrium state of the two fins is not influenced by the added-mass force because the restoring forces acting in the downward and upward directions during the burst period are balanced due to the symmetric profile of Y_avg with respect to the equilibrium position of Y = 0 (see the red line in figure 10b), and the restoring force during the coast period is zero. These results suggest that the asymmetry of the body response by the passive flexibility with respect to the equilibrium positions of the bursting and coasting motions is crucial in determining the distinct physical mechanism in the MTL mode.

Consistent with the observation for the CT mode, the HT mode shows that only circulatory lateral forces without any added-mass lateral force (i.e. zero $\overline {F_y^u}$) influence the equilibrium lateral gap distance between two flexible fins (not shown here) due to the symmetric nature of the body response (figure 4b). Similarly, two rigid fins in the HT mode (Y_head = Y_avg) maintain a lateral stable state due to the balance between the circulatory lateral forces. The current observations from the CT, HT and MTL modes demonstrate that when the lateral motion of two fins is symmetric with respect to an equilibrium position regardless of flexibility and/or intermittency, a lateral equilibrium state is determined by the circulatory lateral forces caused by the enhanced positive pressures and jet deflections. Because we consider only continuous swimming with the symmetric kinematics, the asymmetric kinematics of a continuously swimming fin with respect to the equilibrium position, e.g. prescribed lateral motion with different periods of upstroke and downstroke (Zhao et al. Reference Zhao, Mao, Pan, Huang and Sung2021), most likely leads to the generation of the added-mass lateral force.