4.1. Flight performance on the plane of $\bar {r}_1$ and $Ch$

The flight performance of the wing measured by cycle-averaged lift coefficient $\bar {C}_L$ and power economy $PE$ on the plane of $\bar {r}_1$ and $Ch$ is shown in figure 3, where several interesting results are obtained. Firstly, $\bar {C}_L$ increases with $\bar {r}_1$ when $Ch<0.15$ where the passive pitching is small. This behaviour is similar to the discussion for rigid wings (Ellington Reference Ellington1984b; Shahzad et al. Reference Shahzad, Tian, Young and Lai2016; Bhat et al. Reference Bhat, Zhao, Sheridan, Hourigan and Thompson2019b), which is understandable as the area of the wing is shifted towards the wingtip when $\bar {r}_1$ increases (see also figure 2) generating increasing aerodynamic forces. Secondly, when $Ch$ is large (e.g. ${\ge }0.2$), $\bar {C}_L$ first increases and then decreases with $\bar {r}_1$. The decrease of $\bar {C}_L$ is caused by the large passive pitching and the inertia force during stroke reversal, which have a significant negative effect on lift generation. This observation is consistent with the results obtained with wings modelled by flexible plates of low mass ratio ($M_c\le 1$) and aspect ratio ($AR<3.0$) (Shahzad et al. Reference Shahzad, Tian, Young and Lai2018a,Reference Shahzad, Tian, Young and Laib), and the compliant membrane wings (Gehrke et al. Reference Gehrke, Richeux, Uksul and Mulleners2022; Mathai et al. Reference Mathai, Tzezana, Das and Breuer2022). Such complex behaviour is because flexibility becomes important at large $Ch$, and the FSI system undergoes complex power synchronisation, which will be further discussed in § 4.2. Thirdly, when $Ch$ increases from $0.05$ to $0.6$, $\bar {C}_L$ first increases and then decreases, with the peak located in the band of $0.05\le Ch \le 0.2$. This observation is consistent with the studies with wings modelled by flexible plates (see e.g. Dai et al. Reference Dai, Luo and Doyle2012; Tian et al. Reference Tian, Luo, Song and Lu2013; Shahzad et al. Reference Shahzad, Tian, Young and Lai2018b). Overall, a high $\bar {r}_1$ (${\geq }0.565$) and a low $Ch$ (${\approx }0.1$) is useful in maximising $\bar {C}_L$ with an optimal value of ${\approx }0.67$. Finally, the behaviours of $PE$ in relation to $Ch$ are similar to $\bar {C}_L$, but its maximum value (${\approx }0.71$) is obtained with $\bar {r}_1\approx 0.45$ and $Ch\approx 0.2$. This $\bar {r}_1$ value is very close to that of hawkmoth wings, which has been reported to be $\bar {r}_1\approx 0.47$ (Willmott & Ellington Reference Willmott and Ellington1997). However, in relation to $\bar {r}_1$, $PE$ shows negligible changes for $Ch<0.15$. This will be further investigated in § 4.3.

Figure 3.

Flight performance on the plane of $\bar {r}_1$ and $Ch$: (a) mean lift coefficient $\bar {C}_L$, and (b) power economy $PE$.

We should point out that, even though the above observations are obtained at $Re=300$, $M_c=1.0$ and a specific wing shape, they can be generalised to other conditions (e.g. different $Re$ and $M_c$), with the optimal values and locations dependent on these conditions. This statement is made based on the results from previous studies. Specifically, the optimal $\bar {C}_L$ is larger, but the optimal $PE$ is smaller for higher $AR$, as reported by Shahzad et al. (Reference Shahzad, Tian, Young and Lai2018a), who considered flexible wings at $Re=400$. Moreover, the optimal locations of $\bar {C}_L$ and $PE$ are shifted to larger $\bar {r}_1$ for higher $AR$ (Shahzad et al. Reference Shahzad, Tian, Young and Lai2018a). In addition, insects may fly close to, but not exactly in, the optimal region. For example, the wings of fruit flies are found to be of $\bar {r}_1 \approx 0.53$ (Meng, Liu & Sun Reference Meng, Liu and Sun2017) and $Ch \approx 0.27$ (Lei & Li Reference Lei and Li2020). According to figure 3, $\bar {C}_L=0.43$ might be just sufficient to balance their weight and $PE \approx 0.58$ might be the near-optimal power economy. Overall, the current analysis highlights the strong influence of the coupled effects of wing shape and pitching spring stiffness on wing performance, which will be helpful for designing the flapping-wing flyers and further explain the mechanisms of insect flight.

To summarise this section, two distinct regions of maximum lift production and efficiency are identified. It is found that a high $\bar {r}_1$ (${\geq }0.565$) and a low $Ch$ (${\approx }0.1$) is useful in maximising $\bar {C}_L$, and the power economy is maximised for wings with lower $\bar {r}_1$ (${\approx }0.45$) near $Ch=0.2$.

4.2. Effects of the torsional spring stiffness

To discuss the mechanisms of high lift generation and flight performance, we present the results obtained by varying $Ch$ for three values of $\bar {r}_1$ (0.39, 0.48 and 0.58). The forces, power economy, passive pitching and power components are particularly analysed.

We first discuss $\bar {C}_L$, $\bar {C}_D$ and $PE$, as shown in figure 4. The trends of $\bar {C}_L$ and $PE$ are similar when $Ch$ increases from 0.05 to 0.6, as introduced in § 4.1. The peak values of $\bar {C}_L$ are approximately 0.36, 0.52 and 0.65 for $\bar {r}_1=0.39$, 0.48 and 0.58, respectively (see figure 4a). For all three $\bar {r}_1$ values, the highest value of $\bar {C}_L$ is observed between $Ch=0.1$ and $0.15$, suggesting an optimum range of $Ch$ to achieve the highest possible $\bar {C}_L$. The exact value of the optimum $Ch$ is found to vary within this small range with a change in the wing shape. For all the wings, $\bar {C}_D$ is observed to decrease monotonically with $Ch$, see figure 4(b). This observation is consistent with that of the two-dimensional hovering wing studied, for example, by Yin & Luo (Reference Yin and Luo2010). The decrease in $\bar {C}_D$ for a higher $Ch$ is caused by the larger passive pitching motion resulting in a smaller angle of attack. On the other hand, $PE$ increases with $Ch$, reaching its maximum value at $Ch\approx 0.15$ before decreasing with $Ch$, similar to the trend observed in $\bar {C}_L$ (see figure 4c). From $Ch=0.05$ to $0.15$, $PE$ is increased due to the increase of lift and the decrease of aerodynamic power ($\bar {C}_P$) caused by the softening of the spring. For $Ch>0.15$, $PE$ is found to decrease as the rate of the decrease in $\bar {C}_L$ is higher than that in $\bar {C}_P$. The significant and quick decrease of $\bar {C}_L$ is due to the negative lift caused by the delayed pitch for soft springs and the reduced size of the leading-edge vortex, which will be demonstrated in the analysis of wing kinematics and vortex structures.

Figure 4.

Variations in (a) $\bar {C}_L$, (b) $\bar {C}_D$ and (c) $PE$ are shown as functions of $Ch$ for three different wing shapes of $\bar {r}_1=0.39$, 0.48 and 0.58.

We then present the flapping and pitching power expenditures from various sources, as shown in figure 5, from which several important observations are obtained. Firstly, the flapping aerodynamic power dominates power consumption with peak values roughly 3 times those of the flapping inertial power. This is consistent with the definition of the mass ratio, $M=M_c/AR\approx 1/3$, which indicates that the scale of the inertial forces is approximately 1/3rd of that of the aerodynamic forces. The exception is the case of $Ch=0.6$, where the spring is very soft, resulting in a much smaller aerodynamic force (thus much smaller aerodynamic power) and larger pitching inertial power compared with that of lower $Ch$ cases. Secondly, the flapping aerodynamic power almost always remains positive due to a low mass ratio considered and the continuous work required to overcome the drag. The exception is the case of a very stiff wing (i.e. $Ch=0.05$) with a negative aerodynamic power just before the end of each half-stroke, similar to that observed with a two-dimensional flapping wing in Tian et al. (Reference Tian, Luo, Song and Lu2013), and is caused by the deceleration of the stroke motion. According to two-dimensional simulations (Yin & Luo Reference Yin and Luo2010; Tian et al. Reference Tian, Luo, Song and Lu2013), the cases with high mass ratios could also generate negative aerodynamic power due to a delayed rotation. Normally, the flapping aerodynamic power reaches its maximum value at around midstroke (e.g. $tf=5.25$ and $tf=5.75$), where the drag and the stroke velocity are maximal. Thirdly, for pitching motion, the elastic power of the spring balances the pitching–inertial and pitching–aerodynamic powers and, thus, no net power consumption is observed. Fourthly, the flapping aerodynamic power decreases monotonically with $Ch$ similar to the trend observed in $C_D$ (figure 4b), while the flapping and pitching inertial powers increase monotonically with $Ch$ because the wing of higher $Ch$ travels over a larger $\theta$ range, resulting in higher $\dot \theta$ and $\ddot \theta$. As a result, the pitching elastic power increases correspondingly to balance the increased pitching inertial and aerodynamic powers.

Figure 5.

Time traces of power coefficients ($CP$ values) at $\bar {r}_1=0.48$ for six different spring stiffness values indicated by $Ch=0.05$, $Ch=0.1$, $Ch=0.15$, $Ch=0.2$, $Ch=0.3$ and $Ch=0.6$. $CP_{iner}^{flap}$: flapping inertial power coefficient; $CP_{aero}^{flap}$: flapping aerodynamic power coefficient; $CP_{iner}^{pitch}$: pitching inertial power coefficient; $CP_{aero}^{pitch}$: pitching aerodynamic power coefficient; $CP_{spri}^{pitch}$: pitching elastic power coefficient; $CP_{total}$: total power coefficient. $CP=power/(0.5 \rho U^3 A)$, where $A$ is the surface area of the wing. Here, $tf$ is non-dimensional time, normalised by flapping frequency $f$.

Finally, for $Ch\in [0.1$, $0.2]$, a nearly in-phase synchronisation (with a phase difference ${\rm \Delta} \phi _{flap} \in [18^\circ, 30^\circ ]$ as measured in figure 6a) is observed between the flapping aerodynamic power and the total system input power. Simultaneously, an anti-phase synchronisation (with a mean phase difference ${\rm \Delta} \phi _{pitch} \approx 180^\circ$ as measured in figure 6a) between the pitching elastic power and the pitching aerodynamic and inertial powers is observed. The nearly in-phase synchronisation between the flapping aerodynamic power and the total system input power ensures that the total system input power is mainly used to drive the flapping motion without being wasted in other power components, while the anti-phase synchronisation between the pitching powers ensures that the pitching aerodynamic and inertial powers are fully stored in the elastic pitching power and vice versa. The phase difference between flapping aerodynamic power and the total system input power shows an increasing trend as the increase of $Ch$. The nearly in-phase synchronisation between the flapping aerodynamic power and the total system input power is also reflected by the timing of pitch rotation, which will be further analysed below. As shown in figure 6(b), the maximal absolute value of the peak pitching elastic power occurs at $Ch\in [0.1$, $0.2]$, indicating an efficient storage and release process of the elastic energy in this region. This is consistent with the work by Cai et al. (Reference Cai, Xue, Kolomenskiy, Xu and Liu2022) who reported that the elastic storage minimises the high energetic cost of flapping wings. For $Ch=0.05$, the wing is stiff, and while the synchronisation between flapping and pitching is similar to that required for high lift, the wing's minimal angle of attack is much higher than $45^{\circ }$ where the optimum lift is produced. In addition, the drag is high for $Ch=0.05$ resulting in a low power economy. Wings with $Ch \in [0.1, 0.2]$ produce an optimal lift and a relatively small drag as the wing experiences the minimal angle of attack ${\approx }45^{\circ }$, which will be analysed and discussed in the following parts of this section.

Figure 6.

(a) Phase difference (${\rm \Delta} \phi$) of power coefficients, as a function of $Ch$. Here, ${\rm \Delta} \phi _{flap}$: phase difference between the flapping aerodynamic power and the total system input power; ${\rm \Delta} \phi _{pitch}$: mean phase difference between the pitching elastic power and the pitching aerodynamic and inertial powers. (b) The absolute value of the peak pitching elastic power.

To further study the reasons for high $\bar {C}_L$ and $PE$ with various $Ch$, the time histories of $C_L$, $C_D$, the pitching angle $\theta$ and the corresponding angle of attack $\alpha$ are shown in figure 7; the minimal angle of attack $\alpha _{min}$ and the pitch rotation delay $\delta _{\alpha }$ are shown in figure 8. It should be noted that the pitching angle $\theta$ about the pitching axis is measured with reference to the $z$-axis. Hence, the angle of attack $\alpha$ is defined as

(4.1)\begin{equation} \alpha=\begin{cases} 90^{{\circ}}-\theta, & \text{if $\dot{\phi}<0$},\\ 90^{{\circ}}+\theta, & \text{if $\dot{\phi}\geq 0$}. \end{cases} \end{equation}

Various quasi-steady models correlate $C_L$ and $\alpha$ as $C_L=f[\sin (2\alpha )]$ (Dickinson et al. Reference Dickinson, Lehmann and Sane1999; Sane & Dickinson Reference Sane and Dickinson2001, Reference Sane and Dickinson2002; Wang, Goosen & van Keulen Reference Wang, Goosen and van Keulen2016). Thus, $C_L$ is expected to be maximum when $\alpha \approx 45^\circ$, which is slightly higher than that for a translated flat-plate wing (Taira & Colonius Reference Taira and Colonius2009). On the other hand, $C_D$ is expected to increase with $\alpha$. As the softer springs at higher $Ch$ result in lower $\alpha$ values, the corresponding $C_D$ is also lower, irrespective of $\bar {r}_1$. For a higher $\bar {r}_1$, a larger area is available outboard to support a larger leading-edge vortex (LEV), which results in higher suction magnitudes beneath the LEV. Thus, values of both $C_L$ and $C_D$ are slightly more enhanced in magnitude compared with those at lower $\bar {r}_1$.

Figure 7.

Time traces of $\theta$, $\alpha$, ${C}_{L}$ and $C_D$ are shown for passively pitching wings with four different $Ch$ values. The left column shows the results for the wing of $\bar {r}_1=0.39$ and the right column shows the results for the wing of $\bar {r}_1=0.58$. Data for fruit fly kinematics are obtained from Fry, Sayaman & Dickinson (Reference Fry, Sayaman and Dickinson2005). Here, $tf$ is the non-dimensional time. Note that the results are shown for the 6th flapping cycle at which point the forces and kinematics are periodic. Markers are shown at every 10th time step in the calculations. (a) Time traces of $\theta$ for $\bar {r}_1=0.39$. (b) Time traces of $\theta$ for $\bar {r}_1=0.58$. (c) Time traces of $\alpha$ for $\bar {r}_1=0.39$. (d) Time traces of $\alpha$ for $\bar {r}_1=0.58$. (e) Time traces of $C_L$ for $\bar {r}_1=0.39$. ( f) Time traces of $C_L$ for $\bar {r}_1=0.58$. (g) Time traces of $C_D$ for $\bar {r}_1=0.39$. (h) Time traces of $C_D$ for $\bar {r}_1=0.58$.

Figure 8.

Contours of $\alpha _{min}$ reached in a flapping stroke are shown in (a) on the map of $\bar {r}_1$ and $Ch$, where the white colour represents $\alpha _{ref}\approx 45^{\circ }$. The contours of pitch-rotation delay ($\delta _{\alpha }$) as a percentage of the flapping period are shown in (b) on the map of $\bar {r}_1$ and $Ch$.

The instantaneous values of $C_L$ are also affected by the synchronisation in the flapping and pitching kinematics. The delay in pitch reversal $\delta _{\alpha }$ is calculated as the phase difference between the flapping stroke reversal and the time when $\alpha$ reaches the neutral position of $90^{\circ }$. From figures 7 and 8, a few interesting results are obtained. Pitch reversal is more delayed with increasing $Ch$, as seen in figures 7(a) and 7(b). The delayed pitch causes the instantaneous $\alpha$ after the flapping stroke reversal to be greater than $90^{\circ }$, which results in negative $C_L$ due to the wing being pushed down by the fluid on the wing pressure side. Additionally, the rapid pitch-down motion observed prior to $t/T=5.2$ results in high circulation around the wing that contributes to a significantly high downward force. Thus, $Ch=0.6$ shows negative values of $C_L$ during a part of a stroke, which is responsible for a lower $\bar {C}_L$ of a complete cycle, see figures 7(e) and 7( f). Close to $t/T=5.2$, the pitch-down motion decelerates, followed by the pitch-up motion, which contributes to a positive lift even at low values of $\alpha$ close to $t/T=5.35$. Irrespective of the wing shape, the pitching amplitude of the wing during the midstroke is observed to increase with $Ch$; consequently, the angle of attack $\alpha$ reached near the midstroke is lower, see figures 7(c) and 7(d). The case $Ch=0.15$ experiences the highest instantaneous $C_L$, as the wing nearly maintains $\alpha \approx 45^{\circ }$ (which is also very close to $\alpha _{min}$, see figures 7(c) and 7(e)), which has been shown in previous studies to be the most effective $\alpha$ for lift generation (e.g. Dickinson Reference Dickinson1994; Dickinson et al. Reference Dickinson, Lehmann and Sane1999). The wing of $\bar {r}_1=0.58$ approaches significantly lower values of $\alpha _{min}$. Combined with a delayed pitch, this behaviour results in large negative instantaneous values of $C_L$ as well as an overall lower $\bar {C}_L$. Typically, fruit fly wings have $\bar {r}_1\approx 0.53$ (Meng et al. Reference Meng, Liu and Sun2017) and $Ch\approx 0.27$ (Lei & Li Reference Lei and Li2020). The time traces of $\theta$ and $\alpha$ for a hovering fruit fly from Fry et al. (Reference Fry, Sayaman and Dickinson2005) are compared in figures 7(b) and 7(d) along with a wing with $\bar {r}_1=0.58$. The variations in $\theta$ and $\alpha$ for a fruit fly wing appear to follow the time traces between the cases $Ch=0.15$ and 0.3, which are not in, but close to, the optimal lift region. For all wing shapes, $C_D$ decreases monotonically with $Ch$, caused by the reduced angle of attack for softer springs, see figures 7(g) and 7(h).

The minimal angle of attack $\alpha _{min}$ is found to decrease continuously with increasing $Ch$, see figure 8(a). This is expected since a softer spring will allow the wing to deform more in pitch from its neutral position. Wings with $Ch$ in the range of $[0.1, 0.2]$ produce a maximum $C_L$ (figure 4a) and experience $\alpha _{min}\approx 45^{\circ }$. The value of $Ch$ required to achieve $\alpha _{min}=45^{\circ }$ reduces with $\bar {r}_1$. This is because, for a higher $\bar {r}_1$, the larger area available outboard to support the LEV results in increased suction pressure, whose resultant force makes the wing tilt more than that for a lower $\bar {r}_1$. Consequently, the region of $\alpha _{min}=45^{\circ }$ in figure 8(a) is observed to decrease linearly on the map of $\bar {r}_1$ and $Ch$. This also explains why the region of high $\bar {C}_L$ in figure 3(a) is shifted to lower $Ch$ at high $\bar {r}_1$ values.

The delay in pitch reversal $\delta _{\alpha }$ is seen to increase with $Ch$ for all the examined wing shapes, see figure 8(b). For stiffer springs with $Ch<0.15$, the pitch rotation is advanced with respect to the flapping stroke (i.e. $\delta _{\alpha }<0$), whereas for softer springs, the pitch rotation is delayed (i.e. $\delta _{\alpha }>0$). As has been previously reported by Dickinson et al. (Reference Dickinson, Lehmann and Sane1999) and Sane & Dickinson (Reference Sane and Dickinson2001), the advanced pitch can be useful in achieving higher lift and the delayed pitch may be detrimental. On the other hand, the advanced pitch also results in a higher $C_D$. This confirms the in-phase synchronisation between the flapping aerodynamic power and the total system input power. Similarly, the anti-phase synchronisation between the elastic pitching power and the pitching aerodynamic and inertial powers is reflected in $\delta _{\alpha }\approx 0^{\circ }$ for optimal $PE$ cases (see the $Ch=0.15$ case in figure 5). The region of $\delta _{\alpha }=0$ represents the exact synchronisation between the flapping and pitching motions. This is the region where the highest power economy can be observed for a given wing shape or stiffness, which explains the variations in $PE$ shown in figure 3(b). Note that the cases with high flexibility, i.e. $Ch>0.45$ and $\bar {r}_1>0.6$ have the wing rotated up to and beyond $\alpha _{min}=0^{\circ }$, causing the lift and drag to be highly unstable during a stroke. Overall, $Ch \in [0.1,0.2]$ may be ideal for simultaneously obtaining $\alpha _{min}=45^{\circ }$ to maximise $\bar {C}_L$ and $\delta _{\alpha }\leq 0$ to maximise $PE$.

To summarise this section, the lift production is optimal at $0.05 < Ch \leq 0.2$ and larger $\bar {r}_1$, where the minimum angle of attack is around $45^\circ$ at the midstroke, while the best configuration for the maximum $PE$ is achieved at $\bar {r}_1=0.45$ and $Ch=0.2$, where the flapping aerodynamic power is approximately three times in magnitude of the flapping inertial power.

The optimal performance measured by the power economy is obtained for those cases where two important power synchronisations occur: anti-synchronisation of the pitching elastic power and the pitching aerodynamic and inertial powers and nearly in-phase synchronisation of the flapping aerodynamic power and the total input power of the system.

4.3. Wing-shape effects

The effects of wing shape on the aerodynamic performance are explored by varying $\bar {r}_1$. The variation in the cycle-averaged lift coefficient ($\bar {C}_L$) as a function of $\bar {r}_1$ using three different torsional springs corresponding to $Ch = 0.05$, $0.15$ and $0.45$, is shown in figure 9(a). For the case of the stiffest spring at the wing hinge (i.e. $Ch=0.05$), $\bar {C}_L$ increases monotonically with $\bar {r}_1$. This is in concurrence with previous observations on actively pitching rigid wing shapes with prescribed kinematics. It has been shown that the larger outboard area of a high-$\bar {r}_1$ wing supports the larger LEV that is responsible for creating higher suction on the wing surface, resulting in higher lift (Shahzad et al. Reference Shahzad, Tian, Young and Lai2018a; Bhat et al. Reference Bhat, Zhao, Sheridan, Hourigan and Thompson2019b). However, this observation does not hold true for passively pitching wings with softer torsional springs, see figure 9(a). For wings with softer springs (i.e. $Ch=0.15$ and $Ch=0.45$), $\bar {C}_L$ increases with $\bar {r}_1$ only until a certain $\bar {r}_1$, beyond which it starts decreasing. The value of $\bar {r}_1$ giving a maximum $\bar {C}_L$ changes with $Ch$: $\bar {C}_{Lmax}=0.573$ at $\bar {r}_1=0.565$ for $Ch=0.15$ and $\bar {C}_{Lmax}=0.187$ at $\bar {r}_1=0.48$ for $Ch=0.45$.

Figure 9.

Variations in (a) $\bar {C}_{L}$, (b) $\bar {C}_{D}$ and (c) $PE=\bar {C}_{L}/\bar {C}_{P}$ with $\bar {r}_1$ (wing shape) are shown for passively pitching wings with three different $Ch$ values.

Similar trends are also observed in the cycle-averaged drag coefficient ($\bar {C}_D$), as shown in figure 9(b). For the stiffest hinge (i.e. $Ch=0.05$), $\bar {C}_D$ appears to increase monotonically with $\bar {r}_1$ but for less stiff hinges such as $Ch=0.15$ and $0.45$, $\bar {C}_D$ increases with $\bar {r}_1$ to a maximum after which it decreases with $\bar {r}_1$. Previous studies (e.g. Shahzad et al. Reference Shahzad, Tian, Young and Lai2018a) have shown that, for actively pitching rigid wings, $PE$ decreases with an increase in $\bar {r}_1$ due to an increased $\bar {C}_D$. However, for passively pitching wings, the observations are different, as can be seen in figure 9(c). For $Ch=0.05$, the increase in the outboard area with $\bar {r}_1$, together with a reducing angle of attack $\alpha$ approaching $45^{\circ }$, results in a greater increase in $\bar {C}_L$ than in $\bar {C}_D$, causing $PE$ to increase with $\bar {r}_1$ for $\bar {r}_1>0.43$. However, for $Ch=0.45$, both $\bar {C}_L$ and $\bar {C}_D$ are lower, resulting in the highest $PE$ ($0.466$) at $\bar {r}_1= 0.43$. For the medium-soft spring case ($Ch=0.15$) the balance between spring stiffness and aerodynamic contribution results in the highest $PE=0.662$, which remains virtually constant for $0.43\leq \bar {r}_1 \leq 0.565$.

Overall, the variation in $PE$ appears to directly depend on the pitch-rotation delay caused by the power synchronisations, as discussed earlier. For $Ch=0.05$, the pitch rotation is advanced compared with the flapping motion, as can be seen in figure 8(b). At higher $\bar {r}_1$, the rotation is less advanced, which causes an increase in $PE$ seen here in figure 9(c). At $Ch=0.15$, the delay of $\delta _{\alpha }\approx 2.5\,\%$ is mostly unaffected by $\bar {r}_1$. Thus, the value of $PE$ also remains mostly unchanged. However, at a higher $Ch$, the pitch rotation is more delayed. It first decreases with $\bar {r}_1$, followed by an increase. Hence, the resulting $PE$ also shows a variation as shown for the case of $Ch=0.45$ in figure 9(c).

It can be seen that the choice of the wing shape to achieve the best possible hover performance, both in terms of $\bar {C}_L$ and $PE$, is determined by the spring stiffness used for the passively pitching wings. The reasons behind the trends in $\bar {C}_L$ with various $\bar {r}_1$ are further analysed by comparing their time histories of the pitching angle $\theta$ and the corresponding angle of attack $\alpha$, as shown in figure 10.

Figure 10.

Time traces of the pitching angle $\theta$, angle of attack $\alpha$ and lift coefficient ${C}_{L}$ are shown for passively pitching wings with three different $\bar {r}_1$ values. The left column shows the results for $Ch=0.05$ and the right column shows the results for $Ch=0.45$. Here, $tf$ is non-dimensional time, normalised by flapping frequency $f$. Note that the results are shown for the 6th cycle at which point the forces and kinematics are periodic. Markers are shown at every 10th time step in the calculations. (a) Time traces of $\theta$ for $Ch=0.05$. (b) Time traces of $\theta$ for $Ch=0.45$. (c) Time traces of $\alpha$ for $Ch=0.05$. (d) Time traces of $\alpha$ for $Ch=0.45$. (e) Time traces of $C_L$ for $Ch=0.05$. ( f) Time traces of $C_L$ for $Ch=0.45$.

The pitching amplitude is observed to increase with $\bar {r}_1$, as can be seen in figures 10(a) and 10(b). The corresponding variations in the angle of attack $\alpha$ are shown in figures 10(c) and 10(d). As $\alpha$ approaches $45^{\circ }$, $C_L$ is also observed to increase, in accordance with the quasi-steady models (Dickinson et al. Reference Dickinson, Lehmann and Sane1999; Sane & Dickinson Reference Sane and Dickinson2001, Reference Sane and Dickinson2002), as shown in figures 10(e) and 10( f). On the other hand, for a softer spring, i.e. $Ch=0.45$, the elastic energy from the spring is low and the pitching amplitudes are high, resulting in $\alpha <45^{\circ }$ during the midstroke (e.g. $tf=5.25$). Hence, the corresponding values of $C_L$ at midstroke are lower than those for $Ch=0.05$. Please note that for most cases, $\alpha$ reaches its minimum value before the midstroke and reduces thereafter even though the stroke velocity is still increasing. This is caused by the inertial force of the wing during stroke reversal (see e.g. Dai et al. Reference Dai, Luo and Doyle2012; Tian et al. Reference Tian, Luo, Song and Lu2013). Due to the large pitching motions experienced by the wing with a softer spring, there are also significant changes in the phase of pitch reversal, identified by the phase at which the wing crosses the $\alpha =90^{\circ }$ threshold. Instantaneous $C_L$ can assume significantly negative values near the midstroke for $\bar {r}_1=0.63$ and $Ch=0.45$ (see figure 10f) which contributes to the low $\bar {C}_L$ produced by the wing. The negative $C_L$ is due to the high negative pressure that develops near the trailing edge of the wing pressure side during stroke reversal (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.821 for an animation of the time history of the lift coefficient and the corresponding vortex structures). The animation for $\bar {r}_1=0.63$ and $Ch=0.05$ is also provided for comparison (see supplementary movie 2).

The wing-shape effects can be better understood by observing the flow structures around the wing, as shown in figure 11. Here, the flow over wings of various $\bar {r}_1$ for $Ch=0.05$ and $Ch=0.45$ is shown at the midstroke, i.e. at $tf=5.25$. The LEV, the dominant flow feature shown to be responsible for the high lift produced by flapping wings, is highlighted in all cases. Since the value of $\alpha$ at midstroke is higher with a stiffer spring ($Ch=0.05$) and is closer to 45$^{\circ }$, the size of the LEV over the wings is larger than the LEV that forms over the wing with a softer spring ($Ch=0.45$). The larger LEV creates a higher suction on the wing surface beneath the vortex. Moreover, the LEV is smaller near the root and increases in size in the spanwise direction. Hence, a larger outboard area of the wing with $\bar {r}_1=0.63$ supports a larger LEV, which results in a stronger suction. This explains the increase in $C_L$ with $\bar {r}_1$ for the wing with a stiff hinge (figure 9(a) for $Ch=0.05$) or fully prescribed kinematics (Combes & Daniel Reference Combes and Daniel2001; Shahzad et al. Reference Shahzad, Tian, Young and Lai2018a). For a softer spring with $Ch=0.45$, the suction pressure beneath the LEV increases with $\bar {r}_1$. However, beyond $\bar {r}_1=0.48$, the value of $\alpha$ at the midstroke also reduces significantly, causing the LEV to be smaller and weaker. In addition, the wing of $\bar {r}_1=0.63$ and $Ch=0.45$ rapidly pitches up (i.e. the trailing edge swings up) before $tf=5.2$, leading to a strong vortex at the bottom surface near the trailing edge (see figure 11f), and a strong deceleration occurs thereafter until $tf=5.25$ causing a strong inertia force downwards. Therefore, even with a larger outboard area and a positive angle of attack, the $\bar {r}_1=0.63$ wing produces smaller $\bar {C}_L$, as seen in figure 11( f). Overall, the variation in $C_L$ with $\bar {r}_1$ is found to be dependent largely on the torsional spring stiffness.

Figure 11.

Vortices are visualised by semi-transparent iso-surfaces of the normalised Q-criterion ($Q^{\ast }=100$) and the wing suction sides are coloured by the normalised pressure ($p^{\ast }=p/(\rho U^2)$) contours at the midstroke $tf=5.25$. (a–c) Represent the wings of $\bar {r}_1=0.39$, 0.48 and 0.63, respectively, with $Ch=0.05$.(d–f) Represent the same wings with $Ch=0.45$.

Power expenditures from various sources of four wing shapes $\bar {r}_1=0.39$, $0.48$, $0.58$ and $0.63$ are further analysed in figure 12. For all four wing shapes, the flapping inertial power traces two nearly sinusoidal curves with the positive part being larger than the negative part, see figure 12. The positive flapping inertial power is used to accelerate the wing at the beginning of each stroke cycle (i.e. from $tf=5$ to $tf=5.25$), and the negative inertial power indicates the work done by the fluid on the wing to decelerate it after the midstroke (i.e. from $tf=5.25$ to $tf=5.5$). The pitching inertial power increases with $\bar {r}_1$, due to the significantly increased inertial contribution from flapping (i.e. $J_{zy}$). As a result, the pitching elastic power increases with $\bar {r}_1$ to balance the increased pitching inertial and pitching aerodynamic power.

Figure 12.

Time traces of power coefficients ($CP$ values) at $Ch=0.15$ for four different wing shapes of ${\bar {r}_1=0.39}$, ${\bar {r}_1=0.48}$, $\bar {r}_1=0.58$ and $\bar {r}_1=0.63$. Here, $CP_{iner}^{flap}$: flapping inertial power coefficient; $CP_{aero}^{flap}$: flapping aerodynamic power coefficient; $CP_{iner}^{pitch}$: pitching inertial power coefficient; $CP_{aero}^{pitch}$: pitching aerodynamic power coefficient; $CP_{spri}^{pitch}$: pitching elastic power coefficient; $CP_{total}$: total power coefficient. Power coefficient $CP=power/(0.5 \rho U^3 A)$, where $A$ is the surface area of the wing. Here, $tf$ is non-dimensional time, normalised by flapping frequency $f$.

As $\bar {r}_1$ increases from $0.39$ to $0.58$, the peak value of the flapping aerodynamic power increases by $42.50\,\%$ (i.e. from $1.20$ to $1.71$), followed by a minor decrease ($9.36\,\%$) as $\bar {r}_1$ increases to $0.63$. On the other hand, the flapping inertial power increases monotonically and significantly with $\bar {r}_1$ as the moment of inertia around the flapping axis (i.e. $J_{zz}$) is doubled ($J_{zz}=5.64$ and $11.38$ for $\bar {r}_1=0.39$ and $0.63$, respectively). At $Ch=0.15$, as shown in figure 9, $\bar {C}_D$ and $PE$ are roughly the same for different $\bar {r}_1$, with a $10.15\,\%$ increase of $\bar {C}_L$ for $\bar {r}_1=0.48$ and $\bar {r}_1=0.58$. The slight increase of $\bar {C}_L$ is at the cost of a significant increase in the flapping inertial power ($50.52\,\%$ in the peak value).