3.1 Validation

As mentioned above, the precision-based uncertainty extracted from the specifications of the apparatus and environment was 1.503 %. When applying a basic strategy to control servo motors, however, the use of prescribed motion profiles might have caused additional issues to appear, which would have been difficult to consider (Dickinson et al. Reference Dickinson, Lehmann and Sane1999; Sun & Tang Reference Sun and Tang2002; Birch & Dickinson Reference Birch and Dickinson2003; Lua et al. Reference Lua, Lai, Lim and Yeo2010). In order to settle this question and validate the robotic model, a direct comparison with the result in a previous study was conducted before carrying out the main experiments.

First, the time-varying results obtained from an insect-like flapping wing were identified in order to validate the precision levels of the motions along both axes of the robotic model. The results were then compared with a numerical study (Sun & Tang Reference Sun and Tang2002) at a low Reynolds number ( ${\sim}10^{2}$ ) because it was reasonable to assume that measurement errors derived from mechanical incompleteness, e.g. tolerances or weak rigidity of the parts, affected the accuracy more strongly than errors occurring in previous numerical studies (Sun & Tang Reference Sun and Tang2002; Kweon & Choi Reference Kweon and Choi2010) that did not require turbulence models due to their ultra-low- $Re$ configurations ( $Re\sim 10^{2}$ ). This follow-up experiment was expected to provide a partial explanation for the remaining wake effect issue. (Refer to Birch & Dickinson (Reference Birch and Dickinson2003) regarding the argument for the positive effects of the wake, and Wu & Sun (Reference Wu and Sun2005) for the negative effects. Refer to Kweon & Choi (Reference Kweon and Choi2010) for the different peak levels of the wing–wake interaction.)

All of the properties of this wing were based on the wing model described in the numerical analysis provided by Sun & Tang (Reference Sun and Tang2002). Thus, the wing not only had the same wing planform and aspect ratio, but also the identical second moment of the wing area $\hat{r}_{2}$ . This laser-cut wing was made from 3 mm thick acrylic glass. Angles of $155^{\circ }$ and $100^{\circ }$ were given as $\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D6FC}_{0}$ , and $t/T=0.24$ and 0.32 were employed as the deceleration–acceleration duration and pitching duration respectively. The medium was also changed to a base oil (Super 150N, S-OIL) to create an ultra-low- $Re$ configuration ( $Re\sim 10^{2}$ ).

Figure 4. Plot of $C_{L}$ in a unit wingbeat cycle at low $Re$ .

Figure 4 shows the time-varying $C_{L}$ values in a unit wingbeat cycle, which were obtained using two distinguishable methods, i.e. DNS (Sun & Tang Reference Sun and Tang2002) and the measurements in this study. These two curves show excellent agreement except for a small region right after the stroke reversal. This agreement implies that this robotic model could be controlled more accurately than previous models (Dickinson et al. Reference Dickinson, Lehmann and Sane1999; Han et al. Reference Han, Chang, Kang and Kim2010; Lua et al. 2010), and the results in this paper are sufficient to serve as the basis of validation for future studies. The absence of noticeable peaks right after the stroke reversal indirectly supports the assertions made by Sun & Tang (Reference Sun and Tang2002), who pointed out the detrimental outcome of the wing–wake interaction with respect to lift production (Sun & Tang Reference Sun and Tang2002; Wu & Sun Reference Wu and Sun2005). This result, however, is insufficient to demonstrate the usefulness of the wing–wake interaction and still suggests that further studies are needed. For instance, a topological analysis of the detailed flow structures would enable a better understanding of the wing–wake interaction.

3.2 Underdeveloped wake effects

A number of previous studies on wing–wake interaction (Birch & Dickinson Reference Birch and Dickinson2003; Wu & Sun Reference Wu and Sun2005) based on direct comparisons between the aerodynamic characteristics in the initial stroke and the characteristics in the subsequent stroke have provided insights into the underdeveloped wake in hovering flight. On the contrary, the wake in forward flight has not yet been sufficiently investigated.

Figure 5. Aerodynamic characteristics during three-wingbeat cycles (one forwarding cycle). (a) Time-varying net force coefficients $C_{F}$ in a unit forwarding cycle. (b–d) Mean values: (b) cycle mean $C_{L}$ , (c) cycle mean $C_{D}$ and (d)  $L/D$ .

Figure 5(a) displays the time-varying net force coefficients $C_{F}$ during three-wingbeat cycles in five cases – $J=0.0625$ , 0.1250, 0.1875, 0.25 and 0.5. The trends in each stroke seemed to be identical. As shown here, exactly the same motion profiles were well reflected in each case. For example, the first peaks right after the stroke reversal were found at $t/T\sim 0.07$ and ${\sim}0.57$ , and consistently appeared in every wingbeat cycle (thin black arrows). In the first cycle, however, there was a second peak in the downstroke, which came out at $t/T\sim 0.14$ , and the peak in the upstroke was slightly stronger than the others (thick red arrows). Wu & Sun (Reference Wu and Sun2005) showed that the wing, after several wingbeat cycles, was faced with a downwash, which led to a change in aerodynamic load. Topological research on the vortex behaviour by Han et al. (Reference Han, Chang, Kim and Han2015b ) showed that the force peaks were related to the trailing-edge vortex (TEV) that was generated and shed by the pitching-up wing rotation near the stroke reversal. Thus, it could be interpreted that the lower first peak was caused by a defective TEV, and the second peak and higher $C_{F}$ level originated from the underdeveloped downwash, which was caused by an incomplete wingbeat cycle. At least with respect to the lower- $J$ cases, these inferences are in line with the arguments provided by Sun & Tang (Reference Sun and Tang2002).

Figure 5(b–d) shows the mean $C_{L}$ , $C_{D}$ and $L/D$ in the downstroke phase. In the cases of $J=0.0625$ and 0.1250, the strong $C_{L}$ and $C_{D}$ in the first wingbeat cycle markedly decreased as the cycle progressed, and they converged within the second cycle. The wing in $J=0.1875$ produced moderate variation, but it still had a similar tendency. Such results in the low- $J$ cases showed good agreement with the interpretations by Wu & Sun (Reference Wu and Sun2005), who explained that the fully developed wake reduced the effective angle of attack and the aerodynamic load due to the strong downward flux, i.e. downwash. On the contrary, the aerodynamic loads in two cases – $J=0.25$ and 0.50 – increased slightly along the wingbeat cycle. This implies that the fully developed wake in the higher $J$ was able to assist the aerodynamic force production. An induced flow may have been key to explaining this phenomenon. In the low- $J$ cases, the induced flow in the fully developed wake grew into a downwash, which reduced the effective angle of attack. In the high- $J$ cases, the induced flow may have leaned towards the horizontal direction as the $J$ increased, and this would have led to enhancement of the effective stroke velocity during the downstroke. Such an inversion tendency, however, still requires several more pieces of evidence such as a flow structural diagnosis.

3.3 The coefficients $C_{L}$ , $C_{D}$ and $C_{M}$ , and the behaviour of the centre of pressure

Figure 6. The time-varying (a) $C_{L}$ , (b) $C_{D}$ and (c) $C_{M}$ in a unit wingbeat cycle.

Figure 6 shows the time-varying $C_{L}$ , $C_{D}$ and $C_{M}$ in the last forwarding cycle, i.e. the situation surrounded by the fully developed wake. The $C_{L}$ , $C_{D}$ and $C_{M}$ were extracted using (3.1) and (3.2) respectively, where the reference velocity $U_{ref}$ was defined as $U_{tip,max}\hat{r}_{2}+U_{\infty }$ , i.e. the maximum velocity of the reference point, which was decided by the second moment of the wing area $\hat{r}_{2}$ .

In the case of $J=0$ (hovering), the $C_{L}$ and $C_{D}$ in each stroke had identical variations. Including the two peaks at the start and end of the wing rotation near the stroke reversal, all of these behaviours were safely predictable (e.g. Dickinson et al. Reference Dickinson, Lehmann and Sane1999; Sane Reference Sane2003; Han et al. Reference Han, Chang and Cho2015a ). For example, the peaks before the stroke reversal were identified as rotational circulations, or the ‘Kramer effect’ (Sane & Dickinson Reference Sane and Dickinson2002), and the initial peaks right after the stroke reversal were due to the wing–wake interaction (Sun & Tang Reference Sun and Tang2002; Birch & Dickinson Reference Birch and Dickinson2003; Kweon & Choi Reference Kweon and Choi2010). As the $J$ increased, the aerodynamic loads were gradually weighted towards the downstroke phase, and the curves changed into sinusoidal shapes from the planar tendency of the hovering case. One noticeable feature in the low- $J$ cases was the $C_{L}$ and $C_{D}$ at the middle of the downstroke, which were higher than those in $J=0$ . This implies that a low $J$ can lead to slightly better performance, as revealed by Harbig et al. (Reference Harbig, Sheridan and Thompson2014), who asserted that a moderate $J$ enhanced the vorticity production and induced rapid growth of the LEV.

In the higher- $J$ cases ( $J>0.5$ ), negative $C_{L}$ and $C_{D}$ values were found in the upstroke phase. This may have been because the wing moved to the forward direction even though it was in its upstroke phase. Forward speeds that were faster than the stroke velocity might have led to such values with excessive angles of attack, i.e.  $\unicode[STIX]{x1D6FC}\sim 135^{\circ }$ . Another noticeable feature in the higher- $J$ cases was the linearly subsided $C_{L}$ and $C_{D}$ with a considerable rate in the downstroke phase as the $J$ increased. Bross et al. (Reference Bross, Ozen and Rockwell2013) found that the shed vorticity layer related to the TEV was strongly sensitive to the value of $J$ , which indicated the interference of the lift augmentation. Harbig et al. (Reference Harbig, Sheridan and Thompson2014) also pointed out that the LEV in a higher $J$ with a large $AR$ was eventually shed as part of a vortex loop and drained away to the wing tip. Another noticeable fact was the role of the $J$ in the Navier–Stokes equation (Lentink & Dickinson Reference Lentink and Dickinson2009a ), which was directly able to affect the rotational acceleration numbers. Thus, it could be inferred that the wing in a higher $J$ did not obtain sufficient centripetal and/or Coriolis forces on the wing surface, which are essential in maintaining an LEV and lift augmentation. This inference seems reasonable when considering the results provided by Ozen & Rockwell (Reference Ozen and Rockwell2013), who observed swirl flow degradation and a lifted-off section near the wing tip on the high- $AR$ wing. The gradually disappearing initial peaks also supported the attenuated flow structure around the wing tip, because the wake, which was generated by the previous stroke, locally exerted on the wing surface near the wing tip (Kweon & Choi Reference Kweon and Choi2010).

Similarly to the time-varying $C_{L}$ and $C_{D}$ , the aerodynamic pitching moment coefficient $C_{M}$ in low- $J$ cases barely changed, except at peak levels near each stroke reversal (figure 6 c). This also indirectly implied that moderate $J$ values up to 0.25 would be able to strengthen and stabilize the LEV and consequent aerodynamic loads. Higher peak levels than those in the $C_{L}$ and $C_{D}$ were also explainable via the strong pitching moment during the wing rotation (Han et al. Reference Han, Kim, Chang and Han2015c ). In the higher- $J$ cases, however, the $C_{M}$ in the downstroke phase held at higher levels than those of the lower- $J$ cases. This tendency to act counter to the $C_{L}$ and $C_{D}$ may only be interpreted by changes in underlying causes such as a distorted LEV or other vortical structures that were altered by the adjusted acceleration numbers.

Figure 7. The instantaneous (a) $C_{L_{i}}$ and (b) $C_{D_{i}}$ during the downstroke.

All of the curves changed into sinusoidal shapes, as stated above. In order to determine the underlying causes of this, an additional investigation was carried out on the instantaneous aerodynamic force coefficients $C_{L_{i}}$ and $C_{D_{i}}$ in the downstroke (figure 7 a,b), where the reference velocities in (3.1) and (3.2) simply changed to functions of the stroke profile $\unicode[STIX]{x1D719}(t)$ , hence $U_{inst}(t)=\unicode[STIX]{x1D719}(t)\hat{r}_{2}R+U_{\infty }\cos \unicode[STIX]{x1D719}(t)$ . In the case of $J=0$ , both $C_{L_{i}}$ and $C_{D_{i}}$ in the stroke phase barely changed tendencies. There were small increments along the timeline, but they seemed to be negligible. In other $J$ cases, the wings generated small decrements in their initial stages ( $J<0.25$ ) and moderate peaks near the middle of the stroke, but they were also insignificant. These clearly indicate slight unsteadiness (Han et al. Reference Han, Chang, Kim and Han2015b ) and suggest the appropriateness of the quasi-steady aerodynamic model (Sane & Dickinson Reference Sane and Dickinson2002; Han et al. Reference Han, Kim, Chang and Han2015c ), at least for this higher- $Re$ range. However, these results differed from those of previous studies that showed considerable unsteadiness (Dickson & Dickinson Reference Dickson and Dickinson2004; Harbig et al. Reference Harbig, Sheridan and Thompson2014). It can be assumed that the spanwise flux during the non-perpendicular wing position towards the free stream induced significant effects on the wing at lower $Re$ values ( ${\sim}10^{2}$ ) with strong viscous diffusion (Birch & Dickinson Reference Birch and Dickinson2001; Kim & Gharib Reference Kim and Gharib2010). On the contrary, the strong coherent system near the wing tip with the high-level turbulent characteristics at high $Re$ values ( ${\sim}10^{4}$ ) was able to interfere with the growth of such unsteadiness. Complicated structures including dual and multiple LEVs, burst LEVs and coherent systems at high $Re$ values (Birch et al. Reference Birch, Dickson and Dickinson2004; Lu et al. Reference Lu, Shen and Lai2006; Ansari et al. Reference Ansari, Phillips, Stabler, Wilkins, Zbikowski and Knowles2009; Garmann et al. Reference Garmann, Visbal and Orkwis2013; Johansson et al. Reference Johansson, Engel, Kelber, Heerenbrink and Hedenström2013) seemed to be able to support such inferences.

Figure 8. The behaviours of the centres of pressure during the downstroke ( $0.12<t/T<0.38$ ).

Investigation of the behaviours of the centres of pressure (CPs) can also help to explain the aerodynamic characteristics (Han et al. Reference Han, Kim, Chang and Han2015c ). Figure 8 displays the locations of the CPs during the downstroke, i.e.  $0.12<t/T<0.38$ . These were extracted from two moments along the pitching and stroke axes and the force perpendicular to the wing surface (refer to Han et al. Reference Han, Chang and Kim2014 for the method). All of the CPs were located below the pitching axis, which indicated the pitching-down moment during the stroke. In the non-zero $J$ cases, the CPs were fairly strongly focused on each position. This seems to have been associated with the negligible unsteadiness in the stroke phase. It is of note that such CPs moved from the wing tip to the wing root as the $J$ increased. This also signifies that the increased $J$ may have shrunk the lower-pressure area near the wing tip due to the distorted vortical structure such as the effects brought about by the high $AR$ (Carr et al. Reference Carr, Chen and Ringuette2013; Ozen & Rockwell Reference Ozen and Rockwell2013; Han et al. Reference Han, Chang and Cho2015a ).

The enlarged section in figure 8 represents the CP behaviours in detail. The CPs in each case started at $t/T=0.12$ , as indicated by the solid black markers. In the case of $J=0$ , the CPs were noticeably distributed along the spanwise direction. The initial CP quickly moved from the wing tip and settled down to an inner point within a short period. This implies both the local effect of the wing–wake interaction near the wing tip (Kweon & Choi Reference Kweon and Choi2010; Han et al. Reference Han, Chang, Kim and Han2015b ) and the negligible amount of unsteadiness during the stroke. Small traces of this effect were found in lower- $J$ cases ( $J=0.0625$ and 0.1250), which were in line with the above interpretations for the initial force peaks (figures 6 and 7). In the non-zero- $J$ cases, the CPs moved slightly to the wing root during the stroke. Even the distances of these CP movements seemed to be insignificant, clearly indicating that certain structures near the wing tip, which obstructed the lift production, may have gradually developed during the stroke.

3.4 Understanding flow structures using DPIV

All of the force/moment measurement results can be summarized into a few features. Slight increments in the aerodynamic loads appeared in low- $J$ cases, as they did in a previous study (Harbig et al. Reference Harbig, Sheridan and Thompson2014). The substantial decrements in higher- $J$ cases were also seen in previous studies (Bross et al. Reference Bross, Ozen and Rockwell2013; Harbig et al. Reference Harbig, Sheridan and Thompson2014), as well as in research focusing on the $AR$ (Carr et al. Reference Carr, Chen and Ringuette2013; Ozen & Rockwell Reference Ozen and Rockwell2013; Han et al. Reference Han, Chang and Cho2015a ) with the theoretical description of the Navier–Stokes equation (Lentink & Dickinson Reference Lentink and Dickinson2009a ). The small amount of unsteadiness in all cases was unexpected, but it could have also been elucidated via the high- $Re$ characteristics accompanying the complicated vortical structures around the wing tip (Garmann et al. Reference Garmann, Visbal and Orkwis2013). The CP behaviours (figure 8) indirectly offered information regarding certain obstructions producing the stable LEV attachment, lower-pressure region and the lift augmentation near the wing tip at higher $J$ values. Nonetheless, it was still unclear what the underlying causes were.

Figure 9. The vorticity distributions in the chordwise cross-sections from $0.2b$ to $0.9b$ .

Digital particle image velocimetry was employed to understand these causes. Figure 9(a–f) shows the vorticity fields at the middle of the downstroke, which were taken from the chordwise cross-sections. (See figure 3 for details.) Four cases – $J=0.1250$ and $J=0.1875$ , and $J=0.5$ and $J=1.0$ – were chosen as the lower- and higher- $J$ cases respectively, and all of the vorticity was normalized by the maximum velocity during the downstroke at the reference point as shown in (3.3). Each FOV was reconstructed into three dimensions, and an alpha function was added to uncover the insignificant values. (See the colour bar in figure 9.) Despite these efforts, the quantities along the spanwise direction were not included.

The vortical structures in $J=0$ had very typical structures around a hovering insect wing (figure 9 a). As reported in numerous studies (e.g. Garmann et al. Reference Garmann, Visbal and Orkwis2013), a stable LEV existed on the upper surface of the wing, which grew gradually along the spanwise direction (swirl-like conical LEV; Ellington et al. Reference Ellington, Van den Berg, Willmott and Thomas1996; Van den Berg & Ellington Reference Van den Berg and Ellington1997). The LEV was relatively stretched and dispersed to the trailing edge from $0.7b$ , and the shed vortices were found on the outboard sections of $0.8b$ and $0.9b$ with a slightly lifted-up LEV and rolled-up TEV. It should be noted that this $AR=3$ wing showed the best aerodynamic performance by using the appropriate TV location (Han et al. Reference Han, Chang and Cho2015a ). These structures signify that the TV had grown properly on the outboard sections, which was expressed by both the shed LEV and the TEV.

In the cases of $J=0.1250$ and $J=0.1875$ (figure 9 b,c), the overall vortical formations were nearly identical to that in hover (figure 9 a). Except for the shear layers, the LEVs on the inboard sections seemed to have no differences between them. This resemblance indicates the weak $J$ effect in lower- $J$ cases on the aerodynamic loads (figure 7). With respect to the vorticity in the outboard sections, however, several comparable structures were found. As the $J$ increased to $J=0.1875$ , the LEVs in the sections from $0.5b$ to $0.9b$ were consistently stretched, and they grew without loss of vorticity or distortion. Therefore, the LEVs in the lower- $J$ cases developed into larger conical shapes (Ellington et al. Reference Ellington, Van den Berg, Willmott and Thomas1996), and they occupied a wider range of the wing surface, implying both a lower-pressure area and lift augmentation. These results were similar to those in the research conducted by Harbig et al. (Reference Harbig, Sheridan and Thompson2014), who explained that a non-zero $J$ could enhance the vorticity production at the leading edge. The CPs were located at relatively outer sections in these $J$ cases when compared with the $J=0$ case (figure 8), which also supports this interpretation.

In higher- $J$ cases, the structures were clearly different from those in lower $J$ . In the case of $J=0.5$ (figure 9 d), stretched LEVs were found at inboard sections of $0.4b$ and $0.5b$ with attenuated vorticity. Lifted-up LEVs were also observed at the outboard sections from $0.5b$ to $0.9b$ . One interesting feature was the substantially stretched and degraded LEV, i.e. the LEV dispersions throughout the sections. These dispersions imply that the LEVs were deprived of the concentrated vorticity flux along the spanwise direction, and the wing failed to obtain lift augmentation due to the intensive LEVs. When considering the direction of the vorticity flux and doughnut-shaped vortex ring (Aono, Liang & Liu Reference Aono, Liang and Liu2008), the encroached downwash caused by the incoming TV could also be inferred from the weakened velocities. In the $J=1.0$ case (figure 9 e), the LEVs finally broke down and became irregular despite having similar vorticity levels to those in $J=0.5$ . The TEVs rolled up from the $0.5b$ section, which also indicated a considerable loss of aerodynamic load and a wide TV encroachment. This geometry consisting of defective vortices matched well with the aerodynamic forces in higher- $J$ cases, which showed substantial drops as the $J$ increased (figures 6 and 7). The behaviour of the CPs (figure 8), which were located inside, also aptly support this interpretation.

One noticeable feature of the theoretical interpretation (Lentink & Dickinson Reference Lentink and Dickinson2009a ) is that the $J$ can alter the coefficients of the acceleration numbers similarly to the $AR$ even though the $J$ does not have the same form as the $AR$ in the dimensionless Navier–Stokes equation. Based on the term ( $J^{2}+1$ ), a non-zero $J$ may bring similar outcomes to the effects of an increasing $AR$ . For example, if $J=1.0$ can induce equivalent aerodynamic characteristics to those of twice the $AR$ ( $AR=6$ ), then the radial limit of the stall delay (Kruyt et al. Reference Kruyt, van Heijst, Altshuler and Lentink2015) would be located near $0.5b$ . Figure 9(e) clearly shows that the $0.5b$ section was the border that maintained the LEV without a breakdown. This indicates that at least in terms of LEV stability, the $J$ may have a similar theoretical role to that of the  $AR$ .

Another estimable value is the optimum $AR$ in hover. When considering that the results in lower- $J$ cases did not lead to significant structural changes, converting $J$ may deliver precise $AR$ values to produce optimum aerodynamic loads. If this $AR$ is located between $J=0.1875$ and $J=0.25$ , then the corresponding $AR$ has a range of 3.19 to 3.75 (refer to Lentink & Dickinson (Reference Lentink and Dickinson2009a ) for the equivalent term). However, this estimation should be limited to several configurations, including wing kinematics and the horizontal stroke plane, as well as the Zimmerman wing planform (Shyy et al. Reference Shyy, Aono, Chimakurthi, Trizila, Kang, Cesnik and Liu2010), all of which are different from the configurations in other $AR$ studies (Usherwood & Ellington Reference Usherwood and Ellington2002; Garmann & Visbal Reference Garmann and Visbal2014; Carr et al. Reference Carr, DeVoria and Ringuette2015).

Figure 10. Detailed vortical structures with streamlines at the cross-sections of $0.7b$ .

Figure 10(a–d) describes the streamlines with detailed vortical structures at the $0.7b$ section in four cases – $J=0$ , 0.1875, 0.5 and 1.0. All of the results in figure 10 were also normalized in the same way as those in figure 9. The stronger LEV and longer stretched downwashes in $J=0.1875$ (figure 10 b) adequately explain why the aerodynamic loads were strengthened to a greater degree than those in hover, as Harbig et al. (Reference Harbig, Sheridan and Thompson2014) pointed out. Other particular characteristics were found in higher- $J$ cases. The LEVs in the case of $J=0.5$ were remarkably dispersed to the aft (figure 10 c), with coherent substructures (Hussain Reference Hussain1983; Garmann et al. Reference Garmann, Visbal and Orkwis2013). In $J=1.0$ , the LEV broke down and shed outward (figure 10 d). The rolled-up TEV in higher- $J$ cases should be highlighted. The TEV in $J=0.5$ induced a saddle point near the middle of the chord, and the intense TEV in $J=1.0$ led to an ascending air current in the wake region. These observations were in line with the interpretations given by Bross et al. (Reference Bross, Ozen and Rockwell2013), who described both the insensitive LEV and the delicate TEV depending on the steady incident flow.

It should be noted that the variations of these flow structures could barely be considered close to the effects of the $AR$ (Carr et al. Reference Carr, DeVoria and Ringuette2015; Han et al. Reference Han, Chang and Cho2015a ) or the radius of gyration $r_{g}$ (Wolfinger & Rockwell Reference Wolfinger and Rockwell2014, Reference Wolfinger and Rockwell2015). These effects included a degradation of the spanwise flux and weakened vorticities near the outboard sections as the $AR$ or $r_{g}$ increased. For example, the vortex formation and streamlines around an $AR=6$ wing (Han et al. Reference Han, Chang and Cho2015a ) were much more similar to those around a blunt body, which did not occur in the results for the $J=0.5$ and $J=1.0$ cases in this study. This clearly implies that although the above estimations for LEV stability and lift augmentation were based on the theoretical equilibrium (Lentink & Dickinson Reference Lentink and Dickinson2009a ), these two quantities of $AR$ and $J$ still have to be sorted out. As numerous studies have already revealed (e.g. Carr et al. Reference Carr, Chen and Ringuette2013; Garmann & Visbal Reference Garmann and Visbal2014; Carr et al. Reference Carr, DeVoria and Ringuette2015; Han et al. Reference Han, Chang and Cho2015a ), a high- $AR$ wing did not induce LEV dispersion with disordered structures possessing strong coherent substructures, but reduced the spanwise flux and generated small TVs with small drains. In addition, the strong coherent systems, which were found in the higher- $J$ cases, were one of the characteristics of a low- rather than a high- $AR$ wing (Ozen & Rockwell Reference Ozen and Rockwell2013).

Figure 11. The distributions of the normalized turbulent kinetic energy $\hat{K}$ .

Figure 11(a–d) shows the ensemble averages of the velocity fluctuations along the stroke direction and the turbulence kinetic energy $\hat{K}$ in four cases – $J=0$ , 0.1875, 0.5 and 1.0. Here, the $\hat{K}$ values were normalized by using (3.4) in order to compare them. The results were used to clarify the existence and increments of the turbulent characteristics as the $J$ increased.

In order to validate the turbulent quantities of this paper, we chose several fields in the $J=1.0$ case because it was obvious that the fastest wing had the most white noise and fluctuations in its image. All of the results extracted from the outboard sections had converged within 30 samples, which indicated the appropriateness of the mean and fluctuation values.

The $\hat{K}$ in $J=0$ (figure 11 a) shows typical distributions in hover. There was a small $\hat{K}$ in the root vortex (RV) region, and the $\hat{K}$ was gradually distributed along the drain direction of the spanwise flux, indicating TV development. Of note was the level of $\hat{K}$ in the $J=0.1875$ case (figure 11 b), which was lower than that in the hovering case throughout the fields. When considering that LEV stability implies lift augmentation, the scattered $\hat{K}$ , which was only found in the TV region, may have supported the superior aerodynamic performance in this case (figure 7).

The results in the higher- $J$ cases clearly indicate a randomly distributed $\hat{K}$ , implying unstable vortical fields (figure 11 c,d). In the $J=0.5$ and $J=1.0$ cases, the remarkable growth of $\hat{K}$ started from $0.7b$ and $0.5b$ respectively, and was distributed throughout the outboard sections. The $\hat{K}$ near the leading edge should be noted, which had a weaker $\hat{K}$ than that in the $J=0$ case (figure 11 a). This signifies that the free stream moving over the leading edge did not have a significant impact on LEV stability, at least in areas close to the leading edge (Harbig et al. Reference Harbig, Sheridan and Thompson2014). On the contrary, most of the strong $\hat{K}$ values originated from the trailing edge. These values grew towards to the leading edge as the $J$ increased, and they gradually eroded the flow on the upper surface that had been stable in low- $J$ cases. This $\hat{K}$ behaviour indicates that the high $J$ fed the vortical structure near the outboard trailing edge, and as the $J$ increased, the LEV and substructures were gradually overwhelmed by the unstable TEV. This mechanism was clearly distinct from the $AR$ effects and/or the radius of gyration (Ozen & Rockwell Reference Ozen and Rockwell2013; Wolfinger & Rockwell Reference Wolfinger and Rockwell2014; Han et al. Reference Han, Chang and Cho2015a ).

Figure 12. The near-wake vortical structures at $t/T=0.35$ (during $t/T=0.10$ from the middle of the stroke).

The near-wake structures were also taken via DPIV. Figure 12(a–d) displays the change of near-wake structures in four $J$ cases – $J=0$ , 0.1875, 0.5 and 1.0. These were taken right after the wing passed the middle of the stroke ( $t/T=0.25$ ) to the end of stroke ( $t/T=0.5$ ), and they were reconstructed with respect to the relative distances from the instantaneous wing positions. It should be noted that these do not indicate the instantaneous flow structures at a specific time, but instead reveal time-varying wakes at the middle of the stroke. However, these reconstructions are acceptable regarding the negligible unsteadiness in this study (figure 7).

In the $J=0$ case, the stable RV and TV appeared with the TEV trace (Bross & Rockwell Reference Bross and Rockwell2014, Reference Bross and Rockwell2015). Moreover, the RV and TV that were generated by the previous half-stroke remained around the bottom of the FOV. In $J=0.1875$ , the overall structure (figure 12 b) had identical features to those in $J=0$ . A closer look at this structure, however, indicated slight differences, including a weakened TEV trace and more balanced RV and TV behaviours. This satisfactorily explains the slight improvement in the aerodynamic loads in this case. In higher- $J$ cases, remarkably dispersed RV and TV values, implying unstable vortical structures, were found. The RV in $J=0.5$ (figure 12 c) showed little movement into the wing surface, but the encroachment of the TV seemed to be sufficient to degrade the aerodynamic force generation. The TV in $J=1.0$ (figure 12 d) finally occupied over half of the span, and the RV also encroached into the downwash area. This formation may have accelerated the narrow downwash and the obstruction of the aerodynamic force generation (figures 6 and 7). The TV, which had a similar vorticity level to that of each RV, should also be noticed. The results were clearly distinct from the results of the $AR$ effects (Han et al. Reference Han, Chang and Cho2015a ), which showed substantially degraded TV vorticity on higher- $AR$ wings, as well as the results from Ozen & Rockwell (Reference Ozen and Rockwell2013).

Figure 13. The time-varying downwashes $v^{\ast }$ from $t/T=0.25$ to $t/T=0.5$ .

The time-varying downwashes $v^{\ast }$ from $t/T=0.25$ to $t/T=0.5$ , which were normalized via $U_{ref}$ , were extracted from the line across the half-chord. They were then reconstructed with respect to the relative distances (figure 13 a–f). In three low- $J$ cases – $J=0$ , 0.1250 and 0.1875 – the near wakes were fairly stable. There were downwash decrements right after each trailing edge, signifying the existence of TEVs, but these were small enough to maintain the downwash. The upwashes near each wing tip also indicated TV development, which grew gradually with the increasing $J$ . A noticeable feature was the downwash flux at the farthest fields. The wing in $J=0.1875$ (figure 13 c) had a more balanced distribution than that in hover (figure 13 a). This indirectly implies the better aerodynamic performance of this wing (Han et al. Reference Han, Chang and Cho2015a ), and it also implies that the moderately increased $J$ was able to stabilize the downwash flux as well as enhance the vorticity production of the LEV (Harbig et al. Reference Harbig, Sheridan and Thompson2014). In the $J=0.5$ case (figure 13 d), the TEV traces were substantial. The upwash near the wing tip grew excessively and disturbed the downwash area. In the high- $J$ case (figure 13 e), the near wake developed into highly unstable and dispersed structures. The downwash flux in the far fields also reduced remarkably. Finally, the wing in $J=\infty$ (figure 13 f) generated typical turbulent wake structures, which were similar to structures behind a blunt body or an aerofoil placed in the post-stall region. This clearly implies that the higher $J$ brought on both the downwash decrement and drastic aerodynamic force reduction when the high- $AR$ value was induced (Carr et al. Reference Carr, Chen and Ringuette2013; Han et al. Reference Han, Chang and Cho2015a ). Such high-level turbulent characteristics, however, also indicate that the aforementioned defects in the higher- $J$ cases were due to the deconcentrated vortical structures rather than to the diminished Coriolis and/or centripetal forces (e.g. Han et al. Reference Han, Chang and Cho2015a ).

All of the results for the flow structures clearly indicate that the $J$ is involved in deciding the LEV behaviours to the same extent that the $AR$ decides the LEV behaviours. The optimum $J$ value was close to 0.25, and the aerodynamic forces decreased drastically with increasing $J$ . This is reasonable when considering the role of $J$ terms in the Navier–Stokes equation (Lentink & Dickinson Reference Lentink and Dickinson2009a ) and the $AR$ effects revealed by numerous studies. In addition, the hypothesis that a low $Ro$ stabilizes the LEV (Lentink & Dickinson Reference Lentink and Dickinson2009b ) is reinforced by the results of this study. This is because the $Ro$ can be simplified as $Ro=1+J$ , as Harbig et al. (Reference Harbig, Sheridan and Thompson2014) discussed, and increasing $J$ definitely raises the $Ro$ . However, detailed structures such as the strong upwash, TEV and turbulent characteristics with increasing $J$ were clearly distinct from the structures affected by the $AR$ (Carr et al. Reference Carr, DeVoria and Ringuette2015; Han et al. Reference Han, Chang, Kim and Han2015b ) and/or by the $r_{g}$ (Wolfinger & Rockwell Reference Wolfinger and Rockwell2014, Reference Wolfinger and Rockwell2015), which produced degradation of spanwise flux and weakened vorticities near the outboard sections as the $AR$ or $r_{g}$ increased. This indicates that although the LEV stability and the lift augmentation can be drowned by the rotational force coefficients in the Navier–Stokes equation (Lentink & Dickinson Reference Lentink and Dickinson2009a ), these two different quantities of $AR$ and $J$ have to be separated to understand the underlying mechanisms.

Regarding FWMAV design, the results in this study indirectly provide a flight speed range in a moderate configuration, i.e. $J\sim 0.25$ . This inference garners support when considering fruit flies that maintain a near-horizontal stroke plane during free-flight manoeuvres. (Refer to Ennos (Reference Ennos1989) for Diptera, and Fry, Sayaman & Dickinson (Reference Fry, Sayaman and Dickinson2003) for Drosophila.) Ristroph et al. (Reference Ristroph, Bergou, Guckenheimer, Wang and Cohen2011) showed that the $J$ values of fruit files are distributed within ${\sim}0.3$ . This implies that flight speed might be decided by the range of $J$ as the upper bound being able to stabilize the LEV (figure 9). Moreover, the performance of FWMAVs employing the same control strategy, i.e. asymmetric pitch angles in each stroke for forward flight (e.g. Keennon et al. Reference Keennon, Klingebiel, Won and Andriukov2012), would be confined within this range to avoid LEV shedding and an aerodynamic forces drop. Moreover, it seems that using the inclined stroke plane employed by numerous other insects (e.g.  $J$ of ${\sim}0.8$ in bumblebees, 0.6–0.9 in hawkmoths, ${>}1$ in butterflies; Ellington (Reference Ellington1999)) serves as a method of overcoming this limit and achieving higher speeds. Bross et al. (Reference Bross, Ozen and Rockwell2013) revealed that the LEV was rather stable even at $J>0.5$ when the wing had a stroke plane perpendicular to the flight direction. This most likely occurred because, in contrast to the present study, the $\unicode[STIX]{x1D6FC}$ gradually declined with the increasing free stream. In other words, the shedding of the LEV and the attenuation in the aerodynamic force enhancements would be prevented at a lower $\unicode[STIX]{x1D6FC}$ . However, further research is needed to verify this assertion.

The findings also suggest that the aerodynamic forces and moment estimated by previous aerodynamic models (e.g. Han et al. Reference Han, Kim, Chang and Han2015c ) were inaccurate, except in hover. This is because the models relied on an excessive assumption regarding the stability of the LEV attachment, which would only be rational at $J\leqslant 0.25$ . Fortunately, the results also showed that the quasi-steady assumption was reasonable, at least at this Reynolds number ( $Re\sim 10^{4}$ ) (figure 7). This implies the possibility of building an improved semi-empirical quasi-steady aerodynamic model that allows for an extension to a forward flight configuration.