4.1. Pectoral fin kinematics

To present the variation in kinematics with varied chordwise deformation, we plotted the kinematics of the chord line at $y/SL=0.62$ in figure 5, where the chord line is defined as the straight line connecting the leading edge and the trailing edge at the slice cut. The effective angle of attack can be given by

(4.1)\begin{equation} \alpha_{e} = \tan^{{-}1}\frac{\textrm{d}z/\textrm{d}t}{U}-\theta_p, \end{equation}

where the pitch angle $\theta _p$ is the angle between the inflow speed $U$ and the section chord line, as shown in figure 6$(b)$. Unlike the flapping foil, which has a fixed pitching axis, the heave velocity is defined as $\textrm {d}z/\textrm {d}t$ at approximately the mid-chord point. Figure 6$(a)$ shows that the section motion at $W=0$ is a heaving motion and the chordwise deformation transforms the heaving motion to a combined heaving and pitching motion (figure 6$b$). The study of the 2-D flapping foil has shown that pitching can decrease the effective angle of attack compared with purely heaving (Schouveiler, Hover & Triantafyllou Reference Schouveiler, Hover and Triantafyllou2005a). Similarly, the present transformation results in a slight decrease in heave velocity, and the effective angle of attack decreases significantly as the wavenumber increases.

Figure 5.

Comparisons of instantaneous ($a$) effective angle of attack and ($b$) heave velocity among different wavenumbers for the $y/SL=0.62$ chord line. The chord line is defined as the straight line connecting the leading edge and the trailing edge at the corresponding slice cut.

Figure 6.

Section motion of the $y/SL=0.62$ slice cut at ($a$) $W=0$ and ($b$) $W=0.4$, which can be treated as a heaving and a combined heaving and pitching motion, respectively.

4.2. Hydrodynamic performance

Figure 7 compares the instantaneous force coefficients of the model among the three wavenumbers. The lift force ($F_L$, along the $z$-direction) and the thrust force ($F_T$, along the $x$-direction) are non-dimensionalized by $0.5\rho U^{2} BL^{2}$, with $U$ the inlet velocity, which are denoted as the lift and thrust coefficients, respectively, i.e.

(4.2)\begin{equation} C_{L,T}=\frac{F_{L,T}}{0.5\rho U^{2} BL^{2}}. \end{equation}

Note that the net thrust $F_T$, given by the thrust produced by the motion, $F_x$, minus the drag force $D_p$, i.e.

(4.3)\begin{equation} F_T= F_x-D_p \end{equation}

is usually used to represent the thrust variation (Anderson et al. Reference Anderson, Streitlien, Barrett and Triantafyllou1998; Van Buren et al. Reference Van Buren, Floryan, Wei and Smits2018; Ryu et al. Reference Ryu, Byeon, Lee and Sung2019; Thekkethil, Sharma & Agrawal Reference Thekkethil, Sharma and Agrawal2020a). This definition is more suitable for isolated propulsors, such as oscillating foils and plates. For some animals, we can identify the caudal fin/wing/fluke as the main source of propulsive force for a fish/mammal/bird or insect, and the body as the main source of drag (Liu et al. Reference Liu, Ren, Dong, Akanyeti, Liao and Lauder2017; Han et al. Reference Han, Lauder and Dong2020; Oh et al. Reference Oh, Lee, Park, Choi and Kim2020). Therefore, we can treat the caudal fin/wing/fluke as an isolated propulsor. For the batoid fish, it is hard to distinguish the body and fins; thus, the net thrust of the full-body fish is employed here,

(4.4)\begin{equation} F_T= F_x-D_p-D_{B}, \end{equation}

where $D_{B}$ is the drag of the body. A similar definition was also used in Thekkethil et al. (Reference Thekkethil, Sharma and Agrawal2020b) and for some anguilliform and carangiform swimmers (Maertens, Triantafyllou & Yue Reference Maertens, Triantafyllou and Yue2015; Zhang et al. Reference Zhang, Pan, Chao and Zhang2018). The input power coefficient is defined as

(4.5)\begin{equation} C_P= \frac{P}{0.5\rho U^{3} BL^{2}},P=\int_\varOmega \boldsymbol{F_L} \boldsymbol{\cdot} \boldsymbol{U_{body}}\,\textrm{d}s, \end{equation}

where $\varOmega$ denotes the model surface, $\boldsymbol {F_L}$ is the Lagrangian force and $\boldsymbol {U_{body}}$ is the deformation velocity of the fin at each boundary element $\textrm {d}s$. The swimming efficiency is defined as a quasi-propulsive efficiency (Maertens et al. Reference Maertens, Triantafyllou and Yue2015) according to

(4.6)\begin{equation} \eta=\frac{(\overline{F_T} + R )U}{\bar{P}}=\frac{\overline{C_T}+C_D}{\overline{C_P}}, \end{equation}

where $R$ is the towed resistance at speed $U$ without a swimming motion (i.e. the flat position in figure 1$a$). Here, $C_D$ is the towed resistance coefficient and $C_D=0.1044$ for the present model. To ensure that the flow field can reach the periodic state, the force variation came from the fourth flapping cycle of the simulations.

Figure 7.

Comparisons of instantaneous ($a$) lift, ($b$) thrust, ($c$) total force and ($d$) power coefficients for different wavenumbers. Variations in the cycle-averaged ($e$) thrust coefficient, ($f$) power coefficient and efficiency with the wavenumber.

Figure 7$(a)$ shows that a positive lift force is generated during the downstroke, which is consistent with previous studies on flying animals (Liu et al. Reference Liu, Dong and Li2016; Wang et al. Reference Wang, Ren, Li and Dong2019). A nearly equal and opposite force is generated during the upstroke since all the simulations are conducted at 0 attack angle. Comparing the models with different wavenumbers, a higher lift peak is found at smaller $W$, especially at the mid-downstroke, indicating a lift enhancement for the smaller chordwise deformation. Figure 7$(b)$ shows that the model at $W=0.4$ experiences the largest thrust peak, which can explain why real cownose ray specimens have oscillatory fin behaviours with 0.4 wavenumbers per length (Rosenberger Reference Rosenberger2001). The model with the smallest chordwise deformation ($W=0$) has the smallest thrust peak. We also present the total hydrodynamic force with different deformation in figure 7$(c)$. The smallest chordwise deformation experiences the highest total force. This result is attributed to the lift-based mechanism (circulatory, vorticity) (Bottom et al. Reference Bottom, Borazjani, Blevins and Lauder2016) employed by oscillatory swimmers, in which the total force is created by the flow vorticity around the fins. This is consistent with the larger $\alpha _{e}$ at $W=0$ (figure 5). Similarly, the power consumption at $W=0$ is generally higher than those at larger wavenumbers (figure 7$d$), which indicates that chordwise deformation can decrease the power consumption. These can be better seen quantitatively in the average values of these coefficients and the reason for the force variations are analysed deeply in the following section.

The cycle-averaged coefficients are calculated in table 1. It shows that the swimmer at $W=0$ cannot generate a positive thrust force, i.e. there is not enough net thrust to conquer the drag. The relative thrust enhancement ($\Delta \overline {C_T}$) is computed as

(4.7)\begin{equation} \Delta\overline{C_T} = \frac{\overline{C_T}-\overline{C_T}\mid_{W=0}}{\overline{C_T}\mid_{W=0}+C_D}. \end{equation}

Actually, in (4.7), for the relative difference of thrust, $\overline {C_T}\mid _{W=0}+C_D$ is used. We approximately regard the towed drag as the swimming drag, just as Maertens et al. (Reference Maertens, Triantafyllou and Yue2015) and Schultz & Webb (Reference Schultz and Webb2002) did. On one hand, the body drag $C_D$ avoids a negative value because at $W=0$ the net thrust is a negative value. On the other hand, this is a better presentation of the relative difference excluding the drag force. The relative power and efficiency enhancement are represented as

(4.8)\begin{equation} \Delta\overline{C_P}/\eta = \frac{\overline{C_P}/\eta-\overline{C_P}/\eta\mid_{W=0}}{\overline{C_P}/\eta\mid_{W=0}}. \end{equation}

The thrust is increased by approximately 384 % and 219 % at $W =0.4$ and 0.8, respectively, which suggests a significant hydrodynamic performance enhancement due to the chordwise deformation. The effect of chordwise deformation on the instantaneous force is similar to the effect of wing deformations comprising camber and twist in a dragonfly study (Shumway et al. Reference Shumway, Gabryszuk and Laurence2020). However, the effect on the cycle-averaged force of the cownose ray is much more significant than that of flying animals. In addition, by increasing $W$ from 0.4 to 0.8, the enhancement in thrust decreases, indicating that moderate chordwise deformation is required for optimal thrust performance. To determine the optimal wavenumber, further simulations were performed, as shown in figure 7$(e)$. By increasing the wavenumber, the power consumption decreases almost linearly (figure 7$f$ and table 1), which is consistent with the lower lift force peak for the larger wavenumber. The least power consumption does not result in the highest swimming efficiency because the thrust production shows a peak value at the optimal wavenumber.

Table 1.

Cycle-averaged force coefficients and their enhancements due to chordwise deformation.

4.3. Wake topology

Figures 8 and 9 compare the instantaneous wake structures of cownose ray between different wavenumbers during the stroke cycle. The 3-D vortex structures are visualized by the isosurface of the $Q$-criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). The primary vortical structures include a pair of LEVs, TEVs and tip vortices (TVs). The LEV, shed TEV and TV are connected and form a vortex loop. The wake topology schematic is plotted in figure 10. These types of vortex structures can also be found in previous studies of insects and flapping foils (Liu et al. Reference Liu, Dong and Li2016; Wang et al. Reference Wang, Ren, Li and Dong2019; Thekkethil et al. Reference Thekkethil, Sharma and Agrawal2020b). However, there are some distinct features in the shape and formation of these structures among different wavenumbers. Due to the symmetry between the upstroke and downstroke motions, we discuss only the vortex evolution at the lower surface of the fins.

Figure 8.

Comparisons of the 3-D wake structures of a cownose ray, visualized by the isosurface of $Q=150$ among the models of ($a$–$c$) $W=0$, ($d$–$f$) $W=0.4$, ($g$–$i$) $W=0.8$, at ($a$,$d$,$g$) the early upstroke ($t/T=0.02$), ($b$,$e$,$h$) the mid-upstroke ($t/T=0.22$) and ($c$,$f$,$i$) the late upstroke ($t/T=0.42$).

First, the evolution of the LEVs is presented during the upstroke, as shown in figure 8. At the early upstroke ($t/T=0.02$), shear layers (SLs) are found at the leading edge of the cownose ray's lower surface. There is no significant LEV formed for all three models. At the mid-upstroke ($t/T=0.22$), the SLs are strengthened into LEVs, stably attaching to the lower surface of the leading edge. Around this moment, the LEVs are in the maximum form for all three models, contributing to the force peak, as shown in figure 7. For the smaller chordwise deformation ($W=0$), the LEVs (figure 8$b$) are stronger than those of the other two models (figures 8$e$ and 8$h$) and are formed at almost the whole portion of the leading edge, whereas for the larger chordwise deformation ($W=0.8$), the LEVs are weaker and are formed mainly at the tip of the fins. This difference is due to the change in the effective angle of attack and heaving velocity caused by the chordwise wave (figure 5). At the late upstroke ($t/T=0.42$), the LEVs begin to break from the leading edge, corresponding to the significant force drop at the moment. Meanwhile, significant TVs are developed at the fin tips and form coherent TV tubes.

From the wake structures during the downstroke in figure 9, the TEVs originate from the preceding upstroke. The stroke reversal causes an LEV-to-SL shift at the early downstroke ($t/T=0.52$); i.e. the broken LEVs from the preceding upstroke merge into the negative SL at the middle portion of the fins. At the mid-downstroke ($t/T=0.72$), the SLs are further developed into TEVs and are bounded to the posterior portion of the fins, and then, the TEVs are detached from the trailing edge at the late downstroke ($t/T=0.92$). Due to the stronger LEV for the smaller chordwise deformation, the shed TEVs at this moment become much stronger. Moreover, as seen in figure 9$(c)$ ($W=0$), the two shed TEV tubes from the left and right fins can be seen approaching the centreline of the model and connecting, whereas in figures 9$(f)$ and 9$(i)$, the corresponding TEVs are observed to approach the outboard of the fins.

Figure 9.

Comparisons of the 3-D wake structures of a cownose ray, visualized by the isosurface of $Q=150$ among the models of ($a$–$c$) $W=0$, ($d$–$f$) $W=0.4$ and ($g$–$i$) $W=0.8$, at ($a$,$d$,$g$) the early downstroke ($t/T=0.52$), ($b$,$e$,$h$) the mid-downstroke ($t/T=0.72$) and ($c$,$f$,$i$) the late downstroke ($t/T=0.92$).

Figure 10.

Schematic of the wake structure of the cownose ray model.

4.4. Effects of local vortical structures on the forces

From the above analysis, there is a clear correlation between the force variation and vortex dynamics. The hydrodynamic performance enhancement due to chordwise deformation is a combined effect of the LEV, TEV and TV. To quantify the contribution of these vortical structures to the hydrodynamic forces, DMT (Wu, Lu & Zhuang Reference Wu, Lu and Zhuang2007), a far-field method, was employed here to compute the forces exerted on the boundaries. Unlike the classical near-field method, namely, the integration of the pressure and shear stresses on the solid surface, DMT calculates the hydrodynamic forces directly using the measured velocity field. By altering the integral domain, the effect of various local dynamic processes and structures can be revealed explicitly (Marongiu & Tognaccini Reference Marongiu and Tognaccini2010; Li & Lu Reference Li and Lu2012; Kang et al. Reference Kang, Liu, Su and Wu2018).

For a 3-D incompressible fluid domain $V_f$, surrounding a solid body $B$ and bounded externally by an arbitrary control surface $\varSigma$, the theory developed by Wu et al. (Reference Wu, Lu and Zhuang2007) gives

(4.9)\begin{equation} \boldsymbol{F} ={-}\rho \int_{V_f}\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t}( \boldsymbol{x} \times \boldsymbol{\omega}) \,\textrm{d}V -\rho \int_{V_f} \boldsymbol{l} \,\textrm{d}V -\frac{\rho}{2} \int_{\partial V_f} \boldsymbol{x} \times (\boldsymbol{n} \times \boldsymbol{l})\,\textrm{d}S + \boldsymbol{F_B}+\boldsymbol{F_\varSigma}, \end{equation}

where the domain boundary $\partial V_f$ includes two parts: the boundary of the control surface $\varSigma$ and the body surface $\partial B$; $\partial B$ is deformable and described by the Lagrangian variable, $\boldsymbol {x}$ is the position vector measured from an origin fixed at the model centre, $\boldsymbol {\omega }$ is the vorticity, $\boldsymbol {l}=\boldsymbol {\omega } \times \boldsymbol {u}$ is the Lamb vector, $\boldsymbol {u}$ is the velocity field, $\rho$ is the fluid density, $\mu$ is the fluid viscosity and $\boldsymbol {n}$ is the normal unit of the domain boundary. The fourth term of (4.9) represents the explicit contribution of body acceleration and deformation (Wu et al. Reference Wu, Lu and Zhuang2007). The last boundary integral in (4.9),

(4.10)\begin{equation} {F_\varSigma}={-}\frac{\mu}{2} \int_{\varSigma} (\boldsymbol{x} \times [\boldsymbol{n} \times (\boldsymbol{\nabla} \times \boldsymbol{\omega})])\,\textrm{d}S - \int_{\varSigma}\mu \boldsymbol{\omega} \times \boldsymbol{n}\,\textrm{d}S \end{equation}

expresses the viscous effect on the boundaries.

In our work, the simulations were conducted using the IB method, in which the fluid is filled inside and outside the body boundary $\partial B$ and the $\boldsymbol {n}$ terms on each side of the boundaries counteract each other; thus, the integration in $\partial B$ and the third terms in (4.9) vanish. Meanwhile, the internal fluid inside the boundary is moved with the boundaries, which generates the so-called internal mass effect (Ten Cate et al. Reference Ten Cate, Nieuwstad, Derksen and Van den Akker2002; Suzuki & Inamuro Reference Suzuki and Inamuro2011). For such reasons, DMT theory can be extended as follows for application in the IB method:

(4.11)\begin{equation} \boldsymbol{F} ={-}\rho \int_{V}\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t}( \boldsymbol{x} \times \boldsymbol{\omega})\,\textrm{d}V-\rho \int_{V}\boldsymbol{l}\,\textrm{d}V + \boldsymbol{F_B}'+\boldsymbol{F_\varSigma}', \end{equation}

where $V=V_{Out}+V_{In}$ denotes the whole fluid domain, with $V_{Out}$ and $V_{In}$ representing the fluid domains outside and inside the boundary $\partial B$, respectively, $\boldsymbol {F_B}'$ representing the internal mass effect, i.e.

(4.12)\begin{equation} \boldsymbol{F_B}'=\rho \frac{\textrm{d}}{\textrm{d}t} \int_{V_{In}} \boldsymbol{u}\,\textrm{d}V, \end{equation}

and $\boldsymbol {F_\varSigma }'$ is expressed as

(4.13)\begin{equation} \boldsymbol{F_{\varSigma}}'=\boldsymbol{F_{\varSigma}} -\frac{\rho}{2} \int_{ \varSigma} \boldsymbol{x} \times (\boldsymbol{n} \times \boldsymbol{l})\,\textrm{d}S. \end{equation}

Moreover, Kang et al. (Reference Kang, Liu, Su and Wu2018) found that if a special $\varSigma$ is employed to avoid cutting any discrete vortical structures, all surface integrals over $\varSigma$ disappear. For this reason, the control domain $R$, large enough to contain all the vortical structures, is chosen as the control volume, as shown in figure 11. The hydrodynamic forces calculated by the standard IB method and the improved DMT expression (4.11) are compared in figure 12, which can validate the DMT-based force expression.

Figure 11.

Schematics of control volume $R$ ($R1+R2+R3$) and subdomain divisions in which the LEV, TEV and TV can be captured in $R1$, $R2$ and $R3$, respectively. Only half of the domains are displayed due to the model being symmetric.

Figure 12.

The contribution of subdomains $R1$, $R2$ and $R3$ to the ($a$,$c$,$e$) lift and ($b$,$d$,$f$) thrust at ($a$,$b$) $W=0$, ($c$,$d$) $W=0.4$ and ($e$,$f$) $W=0.8$. The solid black lines and dashed red lines are calculated by standard IB method and DMT expression (4.11), respectively.

To examine the contribution of different vortical structures, the control domain $R$ is further divided into three subdomains $R1$, $R2$ and $R3$ to capture the LEVs, TEVs and TVs, respectively, as shown in figure 11. The integral term of expression (4.11) in each subdomain is compared, as shown in figure 12. The major part of the thrust is associated with the contribution of $R3$, whereas $R1$ and $R2$ mainly make an unfavourable contribution; i.e. the major part of the thrust is contributed by the TV, whereas the LEV and TEV generate more drag. The LEV is the main contributor to the lift force.

At the mid-upstroke ($t/T=0.22$) or mid-downstroke ($t/T=0.72$), when the model experiences a thrust peak, the contribution of the subdomains also reaches its peak. To further understand the hydrodynamic importance of the LEVs and TEVs, we compare the spanwise vorticity ($\omega _y$) contours of several slice cuts through the left fin between three models at the mid-upstroke ($t/T=0.22$). Transparent $Q$-isosurfaces are also displayed associated with the vorticity contours to help identify the vortex structures, in which the LEV and TEV cores are represented by the $\omega _y$ contours overlapping with those identified by $Q$-isosurfaces. As shown in figure 13, stable LEVs exist on the lower surface of fins, growing gradually along the spanwise direction, which can also be observed in a previous study (Garmann et al. Reference Garmann, Visbal and Orkwis2013). It is obvious from the 2-D contours that the LEVs for a small $W$ are much stronger than those for a large $W$. On slice 1, the LEV core structure can be clearly observed (figure 13$a$) for the smallest deformation model, whereas only a slight SL is found for the largest deformation model (figure 13$c$). By comparison, on slices 2, 3 and 4, the LEV cores are still larger in size and occupy a wider range of the fin surface. To further illustrate the effect of chordwise deformation on the LEV, the circulation of the spanwise vorticity is calculated based on the vorticity ($\omega _y$) contours. Figure 14$(a)$ compares the normalized strengths of the LEVs for the three models at the mid-upstroke. The circulation differences give another evidence indicating the reinforcement of the LEV with decreasing $W$. Note that at the fin root (slice 1), the strength of the LEV is near zero for the largest deformation model, coinciding with the LEV tubes identified by the $Q$-isosurface, where the LEV core can grow to a more extended region from the root to tip for the smallest deformation model. Thus, the contribution of $R1$ is highest on both thrust and lift for a smaller deformation due to the stronger LEV.

Figure 13.

Comparison of the instantaneous spanwise vorticity contour ($\omega _y$) among the models of ($a$) $W=0$, ($b$) $W=0.4$ and ($c$) $W=0.8$ at the mid-upstroke ($t/T=0.22$) on four slice cuts (slices 1–4), taken at $0.41SL$, $0.56SL$, $0.69SL$ and $0.83SL$ away from the centreline. Here, $\omega _y$ is normalized by $U/c$, where $c$ is the chord length at $0.62SL$.

Figure 14.

Circulation $\vert \varGamma \vert$ of the ($a$) LEVs and ($b$) TEVs on the four slice cuts at the mid-upstroke ($t/T=0.22$) for the models with different wavenumbers. Here, $\vert \varGamma \vert$ is normalized by $Uc$.

As a result of the LEV–SL–TEV shift, the TEV for the smallest chordwise deformation is similarly stronger (figure 13). On slice 1, a stable TEV can be observed in figure 13$(a)$, whereas it does not even form a TEV core in figure 13$(c)$. As the SL is more developed at the outboard of the fins, a part of the TEV is detached from the trailing edge and forms dual TEVs for the smallest chordwise deformation (from figure 13$(a)$, slices 3 and 4). For the largest deformation, the TEV is attached tightly on the surface and is relatively stretched (figure 13$c$). Similar multi-core vortex structures occur on LEV structures (Lu, Gong & Guo Reference Lu, Gong and Guo2006; Han et al. Reference Han, Chang and Cho2015) due to the high effective attack angle and $Re$, but are not found on TEV structures. Furthermore, we compare the TEV circulation at the mid-upstroke ($t/T=0.22$) for the three models (figure 14$b$). It can be found that at the locations from $0.41SL$ to $0.56SL$, the TEV strength of the smallest chordwise deformation is the highest. As the slice cut moves outboard (at $0.69SL$), the moderate deformation model has the highest TEV strength, indicating that the dual TEVs occur for the least deformation, which weakens the TEV strength since the secondary TEV is detached from the trailing edge. At $0.83SL$, the moderate deformation also generates the dual TEV, and thus, the circulation of the largest deformation is the largest. Therefore, the unfavourable contribution of $R2$ to the thrust decreases slightly, as shown in figure 12$(b)$ compared with figures 12$(d)$ and 12$(f)$.

To illustrate the TV strength, we compare the streamwise vorticity ($\omega _x$) contours of two slice cuts ($x/BL$ = 0.81, 0.92) behind the fins at the mid-upstroke ($t/T=0.22$), as shown in figure 15. For the $W=0$ model (figures 15$a$ and 15$d$), the vortex cores are apparently larger than those of the other models. The vorticity peak values at the vortex cores are approximately $-24.0$ and $-33.9$ for the model in figures 15$(a)$ and 15$(d)$, respectively, whereas they are $-16.0$ and $-11.6$ in figures 15$(c)$ and 15$(f)$, respectively. These values indicate that chordwise deformation can also decrease the strength of TVs. Therefore, subdomain $R3$ of the fins at smaller $W$ contributes more thrust, as shown in figure 12.

Figure 15.

Comparison of the instantaneous streamwise vorticity contour ($\omega _x$) among the models of ($a$,$d$) $W=0$, ($b$,$e$) $W=0.4$ and ($c$,$f$) $W=0.8$ at the mid-upstroke ($t/T=0.22$) on two slice cuts, taken at ($a$,$b$,$e$) $x/BL=0.81$ and ($d$,$e$,$f$) $x/BL=0.92$. Here, $\omega _x$ is normalized by $U/c$, where $c$ is the chord length at $0.62SL$.

Figure 12 (early downstroke) also shows that as the LEV–SL shift progresses during stroke reversal, subdomain $R1$ contributes little thrust. Figure 16 shows the spanwise vorticity contours at $0.41SL$, $0.48SL$, $0.55SL$, $0.62SL$ and $0.69SL$ spanwise slice cuts during the early downstroke ($t/T=0.52$). The cores of the broken LEV are still attached at the lower surface and interact with the SLs. As stated above, the LEV contributes unfavourably to the thrust. For the smallest chordwise deformation in figure 16, the size of the broken LEV cores is much larger, consistent with the stronger LEVs during the LEV–SL shift process, as shown in figure 9. The stronger broken LEVs in subdomain $R1$ contribute more drag; thus, the peak thrust value of the $W=0.8$ model is slightly higher (figure 12$f$) at $t/T=0.52$.

Figure 16.

Comparison of the instantaneous spanwise vorticity contour ($\omega _y$) among the models of ($a$) $W=0$, ($b$) $W=0.4$ and ($c$) $W=0.8$ at the early downstroke ($t/T=0.52$) on five slice cuts (slices 1–5), taken at $0.41SL$, $0.48SL$, $0.55SL$, $0.62SL$ and $0.69SL$ away from the centreline. Here, $\omega _y$ is normalized by $U/c$, where $c$ is the chord length at $0.62SL$.

In our simulations, although the model with the smallest chordwise deformation has the strongest TVs, i.e. that generates the largest thrust in subdomain $R3$, it also has the strongest LEVs, which induce a significant drag. This explains why the model with $W=0$ has the largest total force but the least thrust. In addition, figures 12$(a)$, 12$(c)$, and 12$(e)$ show that the LEV accounts for a large portion of the contribution to lift; thus, the strongest LEV for the smallest chordwise deformation is consistent with the highest lift force. The moderate deformation ($W$ = 0.4) has an optimal balance between the favourable contribution of TV and unfavourable contributions of LEV and TEV, thus generating the highest thrust, as shown in figure 17.

Figure 17.

The cycle-averaged contributions of LEV, TEV and TV to the thrust production. These values are obtained by averaging the contributions of $R1$, $R2$ and $R3$, respectively, from figure 12.

To see more clearly, the thrust is computed as the negative $x$-component of (4.11), i.e.

(4.14)\begin{equation} F_T =\rho \int_{V}\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t}(y\omega_z-z\omega_y)\,\textrm{d}V+ \rho \int_{V}(u_z\omega_y-u_y\omega_z)\,\textrm{d}V- (\boldsymbol{F_B}'+\boldsymbol{F_\varSigma}')_x. \end{equation}

From the first two terms at the right-hand side of the above equation, we can see that the thrust force is related to the distribution of $\omega _y$ and $\omega _z$, while $\omega _x$ does not directly contribute to the thrust force. Figure 18 shows the structures of $\omega _y$ and $\omega _z$ along with the isosurface of $Q$. We can see that the TV subdomain is mainly composed of the $\omega _z$ structure and the $\omega _y$ structure is the main part of the spanwise vortex domain, i.e. the LEV and TEV, at the present condition. Thus the TV and spanwise vortices contribute to the thrust mainly via the information of the $\omega _z$ and $\omega _y$ structures, respectively.

Figure 18.

Comparison of the vortical structures visualized by the isosurface of ($a$) $\lvert \omega _y \rvert =30$ and ($b$) $\lvert \omega _z \rvert =30$ on the model of $W=0.4$, $f=1.0\,\textrm {Hz}$. For orientation purposes, the structures visualized by $Q=150$ as presented in figure 8 are added in a semi-transparent form; thus the $\omega _y$ and $\omega _z$ structures can be considered as the main part of LEV + TEV and TV, respectively.

4.5. Pressure distribution

The pressure distribution of the cownose ray model is presented to further understand the chordwise deformation effects on the hydrodynamic performance. The pressure coefficient is defined as $C_p=(p-p_{in})/0.5 \rho U^{2}$, where $p_{in}$ is the free-stream pressure. Figure 19 shows the pressure contours on the model surfaces at the mid-upstroke ($t/T=0.22$) around which the peaks of thrust and negative lift are observed. On the lower surface, a significant low-pressure region is found, which coincides with the locations of LEVs in figure 8. Due to the higher LEV strength, the model with smaller chordwise deformation has a larger low-pressure region. On the other hand, on the upper surface of the fins, there is a high-pressure region. However, the formation of TEVs leads to a low-pressure region at the back of the upper surface. As the dual TEVs develop more fully and the secondary TEV detaches from the surface, the low-pressure region is relatively smaller in figure 19$(a)$ than that in figures 19$(b)$ and 20$(c)$. The pressure difference between the lower and upper surfaces leads to the hydrodynamic force of the cownose ray. Thus, the higher instantaneous total force generated by the smallest chordwise deformation is attributed to the stronger LEVs and detached TEVs, which enlarge the low-pressure region at the lower surface and reduce the low-pressure region at the upper surface, respectively. Although the smallest chordwise deformation can produce the highest total force, its thrust is the lowest. The reason is that the fin surface is parallel to the swimming direction, which makes the component of the total force in the swimming direction relatively small. At the same time, the total force has a large component in the vertical direction. Moderate chordwise deformation can generate a moderate total force and produce an optimal attack angle, making the force component in the swimming direction the highest.

Figure 19.

Distribution of the pressure coefficient $C_p$ on ($a$–$c$) the upper surface and ($d$–$f$) the lower surface of the model for ($a$,$d$) $W=0$, ($b$,$e$) $W=0.4$ and ($c$,$f$) $W=0.8$ at the mid-upstroke ($t/T=0.22$).

Figure 20.

Comparison of the cycle-averaged contributions of LEV, TEV and TV to the thrust generation for ($a$) the models with $W=0.4$ and different $St$ and ($b$) the models with $St=0.65$ and different wavenumbers. The division of subdomains is the same as that in figure 11.

4.6. Effect of the Strouhal number

The Strouhal number ($St$) is known to be the most important parameter that influences the hydrodynamic performance of aquatic swimmers. To examine the effect of $St$ on the oscillatory swimming of batoid fish, further simulations were performed at lower frequencies $f = 0.7, 0.5, 0.3\,\textrm {Hz}$, which correspond to $St=0.91, 0.65, 0.39$, respectively, as compared with $St=1.30$ at $f=1\,\textrm {Hz}$. It can be observed in figure 20$(a)$ that, with the decrease of $St$, the contributions of local vortical structures decrease; i.e. both the favourable contribution from TV and the unfavourable contribution from LEV and TEV decrease. Note that the thrust from TV decreases to a negative value at $St=0.39$ and the LEV begins to generate positive thrust at $St=0.65$, as the Strouhal number decreases. Figure 21 shows that at $St=0.65$, the LEV region contains more $\omega _z$ structures, while at $St=0.39$, there form very few $\omega _z$ structures in the TV region. Actually, the TV strength is weak at such low $St$ and this observation is consistent with Dong, Mittal & Najjar (Reference Dong, Mittal and Najjar2006). Thus, the difference in the $\omega _z$ structures with those at high $St$ accounts for the favourable contribution of LEV ($St=0.65, 0.39$) and the unfavourable contribution of TV ($St=0.39$).

Figure 21.

The vortical structures visualized by the isosurface of ($a$) $\lvert \omega _z \rvert =10$ (red), $Q=20$ (grey semi-transparent) for the model of $St=0.65$ and ($b$) $\lvert \omega _z \rvert =5$ (red), $Q=5$ (grey semi-transparent) for the model of $St=0.39$.

Furthermore, the cycle-averaged coefficients and their enhancement for different chordwise deformations at $St=0.91$ are shown in table 2. The hydrodynamic enhancements are similar to those at higher $St$. Note that the thrust is enhanced by 315 % at $W=0.4$, whereas it is only 56 % at $W=0.8$. This may be due to the added-mass mechanism at $W=0.8$. It is known that the main parameter in the added-mass mechanism is the so-called slip velocity defined as $U/V$ (Lighthill Reference Lighthill1960), where $V$ is the wave speed ($V=\omega /2{\rm \pi} W$ in this work). A higher slip velocity indicates a higher added-mass force. The slip velocity at $W=0.8$ and $W=0.4$ are 1.14 and 0.57, respectively, at $f=0.7\,\textrm {Hz}$. This indicates that the chordwise deformation can influence the slip velocity, and thus weaken the lift-based force, especially at low $St$. The contributions of local vortical structures to the thrust at $St= 0.65$ are shown in figure 20$(b)$. As expected, the contribution of different vortical structures decreases as the wavenumber increases.

Table 2.

Cycle-averaged force coefficients and their enhancements due to chordwise deformation at $St=0.91$.