3.1 Leading-edge vortex and suction pressure

The LEV structure over a rotating wing is known to depend on the Reynolds number (Lentink & Dickinson Reference Lentink and Dickinson2009). At a higher $Re_{span}$ , the LEV is more compact with a higher spanwise vorticity flux. Since the LEV is responsible for the suction contributing to the overall lift, a change to the LEV structure with $Re_{span}$ can also affect the pressure distribution over the wing surface.

In the present study, the initially square wing was rotated at $Re_{span}=520$ and 6930. The resulting pressure distributions together with the LEV structures are shown in figure 2. High magnitude of suction pressure is present beneath the LEV structures, shown by transparent surfaces of constant $Q$ -criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). Suction is increased at $Re_{span}=6930$ due to the increased surface proximity and spanwise transport of vorticity. At this $Re_{span}$ , the suction increase seems to be very sharp near the LEV boundary, highlighted as location (1) in the figure. However, at $Re_{span}=520$ , the increase in the magnitude of suction is more gradual. From this comparison, it may be inferred that, at a high $Re_{span}$ , the suction is concentrated over a smaller area. At $Re_{span}=6930$ , a small region of vorticity at the wing root is transported diagonally towards the wing tip by the action of the increased Coriolis acceleration.

Figure 2.

Wing suction pressure contours are shown for a wing of $A=1$ rotating at (a) $Re_{span}=520$ and (b) $Re_{span}=6930$ . At a high $Re_{span}$ , the magnitude of suction changes abruptly at location (1). The semi-transparent surface shows the LEV structure identified using the constant $Q$ -criterion. The small black dot represents the location of the wing root.

The net lift on a wing results from the difference between the pressures on the suction ( $p_{s}$ ) and pressure ( $p_{p}$ ) sides. The pressures can be normalised as

(3.1a,b ) $$\begin{eqnarray}\displaystyle p_{s}^{\ast }=p_{s}/(0.5\unicode[STIX]{x1D70C}U_{t}^{2})\quad \text{and}\quad p_{p}^{\ast }=p_{p}/(0.5\unicode[STIX]{x1D70C}U_{t}^{2}). & & \displaystyle\end{eqnarray}$$

The lift coefficient can be improved by maximising the pressure difference averaged over the wing area. This can be achieved by removing wing area where this difference is small. This method of material removal is similar to an evolutionary structural optimisation, such as that described by Xie & Steven (Reference Xie and Steven1997). However, here material was removed based on surface pressure, rather than von Mises stress relevant to structural optimisation. The approach is elaborated in the following section.

3.2 Evolution of wing shapes from pressure contours

Initially, the rectangular wing of $A=1$ was rotated at $Re_{span}=520,1732,3465$ and 6930. The normalised pressures on the wing suction side ( $p_{s}^{\ast }$ ) and the pressure side ( $p_{p}^{\ast }$ ) were extracted. The criterion for material removal was chosen to be the difference $\unicode[STIX]{x0394}p^{\ast }=p_{p}^{\ast }-p_{s}^{\ast }$ , since it contributes to the lift coefficient. The comparison of $p_{s}^{\ast }$ of the four different $Re_{span}$ can be seen in figure 3(a–d), together with the comparison of $p_{p}^{\ast }$ in figure 3(e,f). It is observed that the distribution of $p_{s}^{\ast }$ changes with $Re_{span}$ . However, there is an insignificant change in $p_{p}^{\ast }$ . Note that the pressure acting on the wing edges is neglected as the wing thickness is very small (1 % of $b$ ). The contours of $\unicode[STIX]{x0394}p^{\ast }$ are shown in figure 3(i–l).

Figure 3.

The contours of $\unicode[STIX]{x0394}p^{\ast }$ (bottom row) are obtained by subtracting $p_{s}^{\ast }$ (top row) from $p_{p}^{\ast }$ (middle row) for $Re_{span}=520$ , 1732, 3465 and 6930, in columns one to four, respectively. The black dot represents the position of the wing root and the black horizontal line (at the top of each panel) represents the leading edge. The coloured contours in the last row are superimposed with the black contour lines of constant rejection ratios.

It can be seen that a high $\unicode[STIX]{x0394}p^{\ast }$ is concentrated in the region underneath the LEV, since it creates a large suction. A large area near the wing root and the trailing edge exerts a very low magnitude of pressure. Such inefficiently used material may be eliminated using a criterion for rejection, referred to as the rejection ratio, $RR$ . Sections of the wing surface satisfying the following condition are removed from the model:

(3.2) $$\begin{eqnarray}\displaystyle \frac{{\unicode[STIX]{x0394}p^{\ast }}_{mn}}{{\unicode[STIX]{x0394}p^{\ast }}_{max}}<RR_{i}, & & \displaystyle\end{eqnarray}$$

where the subscript $i$ denotes the design step number, ${\unicode[STIX]{x0394}p^{\ast }}_{mn}$ is the value of $\unicode[STIX]{x0394}p^{\ast }$ at a location of coordinates [ $m,n$ ] on the wing surface and ${\unicode[STIX]{x0394}p^{\ast }}_{max}$ is the maximum value of $\unicode[STIX]{x0394}p^{\ast }$ on the wing surface. The modified wing shapes are rotated at the same Reynolds numbers to evaluate the impact on the lift coefficients. In the subsequent design step ( $i+1$ ), the initial rectangular wing is again cropped using a modified rejection ratio by introducing the evolutionary rate ( $ER$ ), such that

(3.3) $$\begin{eqnarray}\displaystyle RR_{i+1}=RR_{i}+ER,\quad i=1,2,3,\ldots & & \displaystyle\end{eqnarray}$$

In the present study, design steps are performed with the initial rejection ratio $RR_{1}=0.1$ and the evolutionary rate $ER=0.1$ . To provide an overview of the various wing shapes that can be produced at different values of $RR$ , the contours of constant- $RR$ criteria are superimposed onto the colour map of $\unicode[STIX]{x0394}p^{\ast }$ contours in figure 3(i–l). The wing planform in design step $i$ was obtained by cropping the rectangular wing along the contour of ${\unicode[STIX]{x0394}p^{\ast }}_{mn}/{\unicode[STIX]{x0394}p^{\ast }}_{max}=RR_{i}$ . In some cases, such as for $RR_{4}=0.4$ , where the wing root was completely removed, the wingspan was reduced. To maintain a constant wingspan, the wing was provided with an extension using a small strip at the wing root, with a normalised chord $c_{p}/b=0.1$ . In all shapes, the origin was maintained to be at the centre of the chord at the wing root.

3.3 Optimisation of wing shapes using the evolutionary approach

The original rectangular wing shape was modified by creating new planforms using the evolutionary method, as discussed in § 3.2. The modified planforms were also simulated to rotate at the same $Re_{span}$ as that of the rectangular wing. First, for $Re_{span}=520$ , the time traces of the lift ( $L$ ) were extracted and compared, as shown in figure 4. Here, the rejection ratio $RR_{0}=0$ corresponds to the original rectangular planform. It can be seen that $L$ reaches a peak during the initial acceleration and then drops. Furthermore, after the rotation of approximately $100^{\circ }$ , $L$ reaches a stable value due to the stable LEV, as also shown by Birch et al. (Reference Birch, Dickson and Dickinson2004) and Carr et al. (Reference Carr, Devoria and Ringuette2015). Hence, the mean lift ( $\overline{L}$ ) can be obtained by averaging over the final $30^{\circ }$ of rotation. It can be seen that $\overline{L}$ reduces with $RR$ due to a reduction in wing area. This reduction can be represented by the relative lift, $\overline{L}/\overline{L}_{0}$ , where $\overline{L}_{0}$ is the mean lift over the rectangular wing at the same $Re_{span}$ . Similarly, the wing area ( $S$ ) relative to that of the rectangular wing ( $S_{0}$ ) can be represented by $S/S_{0}$ .

Figure 4.

The time traces of $L$ are shown for the wing planforms obtained by rejection ratios $RR_{2}=0.2$ (red), $RR_{3}=0.3$ (green) and $RR_{4}=0.4$ (blue), rotating at $Re_{span}=520$ . The ratio $RR_{0}$ (black) corresponds to the original rectangular planform.

It can be seen in figure 5(a) that, even though $\overline{L}/\overline{L}_{0}$ on the wing is reduced with $RR$ , the relative area, $S/S_{0}$ , is reduced significantly more. This results in an improved mean lift coefficient, calculated as

(3.4) $$\begin{eqnarray}\displaystyle \overline{C}_{L}=\frac{\overline{L}}{0.5\unicode[STIX]{x1D70C}U_{t}^{2}S}. & & \displaystyle\end{eqnarray}$$

Here, the quantities $\unicode[STIX]{x1D70C}$ and $U_{t}$ are maintained to be constant. Hence, $\overline{C}_{L}$ can be considered as a non-dimensional representation of $\overline{L}/S$ . As shown in figure 5(b), the value of $\overline{C}_{L}$ increases with $RR$ , with the peak reached at $RR=0.3$ . Beyond $RR=0.3$ , $\overline{C}_{L}$ starts decreasing, indicating that $RR=0.3$ gives the maximum lift coefficient at $Re_{span}=520$ . Note that, in the evolutionary structural optimisation, the material is removed iteratively at any given $RR_{i}$ , until the steady state is reached. On the contrary, in this evolutionary approach, only one cropping iteration is performed at a given $RR_{i}$ , since the pressures on the wings will get readjusted after rotating the cropped wing. After any iteration, there will always be some part of the wing area satisfying the condition in (3.2), and hence no steady state can be reached. Moreover, the incremental material removal at every iteration will require significantly more design changes and simulations, resulting in considerably more expensive computations. An example of such iterative process at $Re_{span}=520$ is shown in figure 5(b). Here, the wing has been cropped iteratively at the steps $RR=0.1$ and $RR=0.2$ . The reduction in the wing area relative to the original rectangular wing area ( $S_{0}$ ) is shown by the quantity $\unicode[STIX]{x0394}S^{\ast }=(S-S_{0})/S_{0}$ . The improvement in $\overline{C}_{L}$ for $RR=0.1$ follows a similar trend, initially, as in the non-iterative process, followed by a reduction in $\overline{C}_{L}$ . Furthermore, for $RR=0.2$ , the improvement in $\overline{C}_{L}$ is less than 3 %. All the $\overline{C}_{L}$ values from these two $RR$ criteria are lower than $RR=0.3$ . Hence, the maximum value of $\overline{C}_{L}$ is indeed at $RR=0.3$ , which can be obtained even with the faster non-iterative process.

Figure 5.

(a) At $Re_{span}=520$ , the stepwise design variations of the mean lift relative to that of the rectangular wing ( $\overline{L}/\overline{L}_{0}$ ), the wing areas relative to that of the rectangular wing ( $S/S_{0}$ ) and the normalised radii of the second moment of area ( $\hat{r}_{2}$ ). (b) The variation of the mean lift coefficient ( $\overline{C}_{L}$ ). The variations in $\overline{C}_{L}$ using the iterative cropping method at $RR=0.1$ and 0.2 have also been compared. (c) The spanwise variations in the sectional lift coefficient ( $\text{d}\overline{C}_{L}/\text{d}r^{\ast }$ ) for various $RR$ .

The improvement in $\overline{C}_{L}$ with $RR$ can be attributed to the change in the spanwise distribution of pressure, which can be observed by evaluating the sectional lift coefficient ( $\text{d}\overline{C}_{L}/\text{d}r^{\ast }$ ) along the wingspan. Here, $r^{\ast }$ represents the radial location of a wing section normalised by the wingspan. Coefficient $\text{d}\overline{C}_{L}/\text{d}r^{\ast }$ can be obtained at a spanwise section by integrating $\unicode[STIX]{x0394}p^{\ast }$ along the chord at that section. The spanwise variations in $\text{d}\overline{C}_{L}/\text{d}r^{\ast }$ for various values of $RR$ are compared in figure 5(c). It can be seen that, with an increase in $RR$ , $\text{d}\overline{C}_{L}/\text{d}r^{\ast }$ reduces inboard and increases significantly outboard. The peak value of $\text{d}\overline{C}_{L}/\text{d}r^{\ast }$ , seen roughly near $r^{\ast }=0.9$ , decreases beyond $RR=0.4$ . Although the peak value for $RR=0.4$ is slightly higher than that for $RR=0.3$ , the overall variation in $\text{d}\overline{C}_{L}/\text{d}r^{\ast }$ at $RR=0.4$ shows lower values throughout the span, except in the range $0.8<r^{\ast }<0.95$ . Therefore, $\overline{C}_{L}$ obtained for $RR=0.3$ is the highest.

In general, the modified planforms obtained from the design evolution were found to have a significantly greater $\overline{C}_{L}$ than the rectangular wing. This process was repeated for different $Re_{span}$ and the values of $\overline{C}_{L}$ were computed to determine the optimum $RR$ . The variation in $\overline{C}_{L}$ with $RR$ for various $Re_{span}$ is shown as a contour map in figure 6(a). It can be seen that, for any $Re_{span}$ , $\overline{C}_{L}$ reaches a maximum value at $RR=0.3$ and starts decreasing beyond this ratio. Hence, optimised wing shapes can be obtained with $RR=0.3$ across all Reynolds numbers. The values of $\overline{C}_{L}$ for these optimised wing shapes are plotted as a function of $Re_{span}$ in figure 6(b). The comparison with the values for the original rectangular wing at the same $Re_{span}$ shows a remarkable 40 % improvement in $\overline{C}_{L}$ . Moreover, the rotations of a fruit fly wing and of a bumblebee forewing were also simulated at various $Re_{span}$ to compare their $\overline{C}_{L}$ values. It should be noted that the geometrical parameters of the optimised wings are different from those observed in nature, since there might be several other factors affecting the wing morphology, as discussed later. In fact, although the $A$ values of the optimised wings do not match with those of the insect wings, they are within the optimum range of $A$ at the respective span-based Reynolds numbers shown in our recent work (Bhat et al. Reference Bhat, Zhao, Sheridan, Hourigan and Thompson2019).

Figure 6.

(a) The contours of $\overline{C}_{L}$ plotted on the plane of $RR$ and $Re_{span}$ . (b) The variation of $\overline{C}_{L}$ plotted against $Re_{span}$ for the rectangular wing, optimised shapes at $RR=0.3$ and planforms of a fruit fly wing and a bumblebee wing. The optimised $\overline{C}_{L}$ values are highlighted with black dots in (a) and the corresponding shapes are shown in grey colour in (b).

Table 1.

Characteristics of the wing shapes.

From figure 6(b), it is clear that the optimised wings have more area outboard. Hence, their centroids are further away from the wing root compared to a rectangular wing. The normalised radii of the first and second moment of inertia, i.e. $\hat{r}_{1}$ and $\hat{r}_{2}$ , respectively, for various wings are summarised in table 1. This shows that the wings with a higher $\hat{r}_{1}$ have a higher $\overline{C}_{L}$ . Interestingly, forewings of insects are found to have a larger area outboard and a recent study by Chen et al. (Reference Chen, Kolomenskiy, Nakata and Liu2018) shows that these wings provide most of the lift, whereas the hindwings mostly aid in manoeuvrability. The values of $\hat{r}_{1}$ and $\hat{r}_{2}$ for the bumblebee forewing are higher than those for the rectangular wing and the fruit fly wing and lower than those for the optimised wings and so is its $\overline{C}_{L}$ , consistent with the findings of Shahzad et al. (Reference Shahzad, Tian, Young and Lai2016). Therefore, in this case, $\overline{C}_{L}$ can be considered to be artificially increased, since a larger percentage of the area of the cropped wings is subjected to higher velocities.

However, it must be noted that this relation between $\hat{r}_{1}$ , $\hat{r}_{2}$ and $\overline{C}_{L}$ is not always the same. For example, as shown in figure 5, the value of $\hat{r}_{2}$ , for the wing shapes derived by increasing $RR$ , increases. However, the value of $\overline{C}_{L}$ is decreased beyond $RR=0.3$ , even if its $\hat{r}_{2}$ is increased. This is because $\overline{C}_{L}$ is affected not only by $\hat{r}_{2}$ , but also by the change to the mean wing-surface pressure caused by the proximity of the trailing edge vortex after cropping. The increase in $\hat{r}_{2}$ contributes to an increase in $\overline{C}_{L}$ , whereas the proximity of the opposite sign vorticity from the trailing edge contributes to a decrease in the magnitude of the mean surface pressure, and hence to a decrease in $\overline{C}_{L}$ . The distance $d_{LT}$ between the LEV and the trailing edge at a certain spanwise location can be obtained, as shown in figure 7(a). Here, the LEV has been identified using the constant $Q$ -criterion and the spanwise location is chosen to be at $\hat{r}_{2}$ to allow comparison across various wing shapes. It can be seen in figure 7(b) that $d_{LT}/b$ reduces at the first design step and remains relatively stable until $RR=0.3$ . On the other hand, there is an increase in $\hat{r}_{2}$ with $RR$ , which would increase $\overline{C}_{L}$ . Beyond $RR=0.3$ , $d_{LT}/b$ is observed to drop while having no significant change to $\hat{r}_{2}$ . Thus, $\overline{C}_{L}$ is expected to decrease beyond $RR=0.3$ , which is consistent with our observations. Therefore, it is important to derive a wing profile such that the counteracting effects of $\hat{r}_{2}$ and trailing edge proximity are optimised to achieve the maximum $\overline{C}_{L}$ . The proposed evolutionary approach demonstrates the derivation of such optimum wing profile.

Figure 7.

(a) The minimum distance ( $d_{LT}$ ) between the LEV, shown by the blue contour, and the trailing edge (TE), is calculated at a spanwise section at $\hat{r}_{2}$ . Here, the black line represents the wing cross-section with the leading edge shown by a filled circle and the trailing edge shown by a hollow circle. (b) The variation of the normalised distance $d_{LT}/b$ with changing $RR$ for wings rotating at $Re_{span}=520$ .

As can be seen in figure 8, the wing-surface pressures are distributed more efficiently on the optimised wings than on a rectangular wing. At $Re_{span}=1732$ , 3465 and 6930, it can be seen that the wing shapes are just sufficient to support the LEV structures. Moreover, when compared to a rectangular wing, the magnitude of the suction pressure for the optimised wings has been slightly reduced by the action of the opposite sign vorticity from the trailing edge. This is because the trailing edge has moved closer to the leading edge in the inboard area of the wing. However, this reduced magnitude is insignificant compared to the increase in $\overline{C}_{L}$ from eliminating a large amount of low-performing area. Furthermore, unlike for rectangular wings, a small amount of trailing-edge vorticity is also observed to tilt into the wake at a location of approximately 30 % of the span of the optimised wings. This may be due to the inability of the trailing-edge vortex to advect the vorticity along the trailing edge beyond a certain point. The trailing edge becomes nearly parallel to the chord at the midspan location, requiring the advection of the trailing-edge vorticity along the chordwise direction. The separation of this vorticity from the trailing edge might affect the suppression of the suction pressure, restoring a higher $\overline{C}_{L}$ .

Figure 8.

Normalised surface pressures on the optimised shapes ( $RR=0.3$ ) are distributed efficiently at (a) $Re_{span}=520$ , (b) $Re_{span}=1732$ , (c) $Re_{span}=3465$ and (d) $Re_{span}=6930$ . The LEV structures are represented by the semi-transparent isosurfaces of constant $Q$ -criterion.

Some studies (Lee, Lua & Lim Reference Lee, Lua and Lim2016; Jardin & Colonius Reference Jardin and Colonius2018) indicate that $\overline{C}_{L}$ is dependent on both the wing aspect ratio and Rossby number. Our recent study (Bhat et al. Reference Bhat, Zhao, Sheridan, Hourigan and Thompson2019) has shown that, at a constant span-based Reynolds number, $\overline{C}_{L}$ decreases with an increase in $A$ . Also, $\overline{C}_{L}$ has been shown to decrease with an increase in the span-based Rossby number $Ro_{b}=r_{2}/b=\hat{r}_{2}$ . In the present case, as the wing is cropped, reducing the mean chord, $A$ and $\hat{r}_{2}$ are both simultaneously increased, which can cause a decrease in $\overline{C}_{L}$ . However, with a constant rotation rate, the velocity at the radius of gyration is also increased, which can increase $\overline{C}_{L}$ . Therefore, a combined effect of these factors is observed, which might provide an additional explanation to the observed variation in $\overline{C}_{L}$ with $RR$ .

3.4 Effects on power economy

Although the main optimised parameter was the lift coefficient, or specifically the lift force per unit area, the impact of the derived optimal shapes on the power economy ( $PE$ ) was also investigated. The power economy is defined as the ratio of the mean lift coefficient and the mean power coefficient ( $PE=\overline{C}_{L}/\overline{C}_{P}$ ), where the mean power coefficient is $\overline{C}_{P}=(\overline{\unicode[STIX]{x1D70F}}_{y}\unicode[STIX]{x1D6FA})/(0.5\unicode[STIX]{x1D70C}U_{t}^{3}S)$ . Here, $\overline{\unicode[STIX]{x1D70F}}_{y}$ is the mean fluid mechanical torque acting along the axis of rotation and $\unicode[STIX]{x1D6FA}$ is the constant angular velocity. The comparison of the values of $PE$ for the optimised wing shapes at different Reynolds numbers with those for the rectangular wing is shown in figure 9(a). This demonstrates that the power economy of the optimised shapes is lower for any given Reynolds number than that for the rectangular wing. It is important to note that the improvement in $\overline{C}_{L}$ of the optimised wing shapes is huge ( ${\sim}50\,\%$ ) compared to the reduction in their $PE$ ( ${\sim}15\,\%$ ). The reduction in $PE$ with increasing $\hat{r}_{1}$ is also in accordance with the results of Shahzad et al. (Reference Shahzad, Tian, Young and Lai2016). Interestingly, the fruit fly wing is observed to have $PE$ higher than that of the rectangular wing, even while having a similar $\hat{r}_{1}$ . The reasons behind this difference were investigated further.

Figure 9.

The variations of (a) power economy ( $PE$ ), (b) $\overline{C}_{L}/\overline{C}_{D}$ and (c) normalised location of the point of application of drag ( $\hat{r}_{D}$ ) with $Re_{span}$ are shown for the rectangular wing, optimised wing shapes and the fruit fly wing.

Wings with lower power economy require more power to overcome the mean torque $\overline{\unicode[STIX]{x1D70F}}_{y}$ . This torque is derived from the mean drag coefficient ( $\overline{C}_{D}$ ) and the radial location of the centre of pressure. Here, both $\overline{C}_{L}$ and $\overline{C}_{D}$ directly depend on the wing-surface pressures. However, the ratio $\overline{C}_{L}/\overline{C}_{D}$ increases with Reynolds number (Harbig et al. Reference Harbig, Sheridan and Thompson2013), which might be responsible for the increase in $PE$ with $Re_{span}$ . To observe the effect of wing shape, the variation of $\overline{C}_{L}/\overline{C}_{D}$ with $Re_{span}$ was plotted for different wings, as shown in figure 9(b). Although the optimised wings have lower $\overline{C}_{L}/\overline{C}_{D}$ than the rectangular wing, the difference is less than 5 %. This implies that the increase in $\overline{C}_{L}$ of the optimised wings is accompanied by an increase in $\overline{C}_{D}$ by nearly an equal amount. Therefore, the significant contribution to the change in $PE$ must be from the change in the moment arm. Hence, the variation of the normalised location of the point of application of drag ( $\hat{r}_{D}=r_{D}/b$ ) was plotted for different wings, as shown in figure 9(c). The optimised wings have higher $\hat{r}_{D}$ , which explains the increase in $\overline{C}_{P}$ . The fruit fly wing has the lowest $\hat{r}_{D}$ , and therefore the highest $PE$ . Decreasing $\hat{r}_{1}$ can further decrease $\hat{r}_{D}$ to obtain even higher $PE$ . Insect wings could have evolved to their present shapes as a result of a compromise between maximum lift coefficient and maximum power economy. There may be several other factors, such as the resistance to bending and resistance to torsion, which require a broader chord inboard (Ennos Reference Ennos1989; Wootton Reference Wootton1992). The evolution of insect wing morphology has also been affected by environmental factors and the requirement for thermo-regulation, as has been discussed by Kingslover & Koehl (Reference Kingslover and Koehl1994) and Johansson, Söderquist & Bokma (Reference Johansson, Söderquist and Bokma2009). It may be interesting to note that the forewings of insects like bumblebees have a larger area outboard, i.e. a high value of $\hat{r}_{1}$ , and their integral wings (forewing and hindwing together) have a lower $\hat{r}_{1}$ (Chen et al. Reference Chen, Kolomenskiy, Nakata and Liu2018).

Overall, it was observed that a higher $\overline{C}_{L}$ would require a larger area outboard. However, the value of $\overline{C}_{D}$ would also increase simultaneously, as both the values depend on pressure. Hence, the power requirement to overcome the drag would also increase. One method of reducing the drag is to lower the angle of attack. However, it has been shown by Kruyt et al. (Reference Kruyt, van Heijst, Altshuler and Lentink2015) that lower angles are beneficial for the wings of very large $A$ (and at a high $Re_{span}$ ). This is because, at a low $Re_{span}$ , lower angles of attack may create a smaller LEV, which would create a lower suction. Hence, the lift coefficient would also decrease, which is not desirable.

Another method of reducing the power requirement is to lower the inertial torque that scales with $\hat{r}_{2}$ . For a simple flat plate planform, the radius of the second moment of area, $\hat{r}_{2}(S)$ , is the same as that of the radius of mass moment of inertia, $\hat{r}_{2}(m)$ . However, $\hat{r}_{2}(m)$ can be changed from $\hat{r}_{2}(S)$ by using a non-uniform thickness of the wing. Accordingly, the centre of mass can be shifted inward for the same planform by adding more mass to its root. The centre of pressure, which affects $\overline{C}_{L}$ and $\overline{C}_{D}$ , is dependent on the spanwise wing area distribution, as seen in this study. Therefore, a larger $\overline{C}_{L}$ can be obtained by more outwards wing area. Simultaneously, a lower inertial torque will be required to rotate or flap the wing when the centre of mass is moved inward using a thicker wing root. Interestingly, it can be seen that insect wings have a thick axillary area at the wing root, where all the wing venation is connected. The veins are thinner towards the tip, helping $\hat{r}_{2}(m)$ to be located more inward. Such design may help the wings to have a larger $\overline{C}_{L}$ and a lower $\overline{C}_{P}$ simultaneously.

Autorotating samaras do not require an internal power. The force driving their motion is provided by gravity. However, they require some additional features, such as a spanwise twist in the wing, to be able to autorotate. The twist helps in initiating the autorotation by providing an asymmetric resistance to the air flow when the seed starts falling (Norberg Reference Norberg1973). It also creates a uniform induced flow along the span, which is beneficial for the uniform distribution of the aerodynamic forces (Low et al. Reference Low, Pheh and Foong2016). Moreover, the leading edge of a samara wing is tilted slightly down at its root. The angle made by the leading edge with the horizontal plane is called the coning angle, $\unicode[STIX]{x1D6FD}$ . A low and critical value of $\unicode[STIX]{x1D6FD}$ is maintained to be constant when the low, radially inward aerodynamic force is balanced by the centrifugal force (Norberg Reference Norberg1973). At a constant rotational velocity, the low centrifugal force can be achieved by adjusting the location of the centre of mass, $\hat{r}_{1}(m)$ , radially inward. Indeed, in samaras, the heavy seed at the root of its lightweight wing helps in maintaining the centre of mass close to the root ( $0.1<\hat{r}_{1}(m)<0.2$ ).

However, as mentioned earlier, the focus of the present study is the improvement in $\overline{C}_{L}$ , which is shown to be significantly higher for the optimised shapes. The relatively small reduction in $PE$ is noted as a secondary effect of this process.