Dual leading-edge vortices on flapping wings

The leading-edge vortex (LEV), a flow structure responsible for the high-lift generation in insect flight, is well known to both biologists and fluid-mechanists, and has been widely studied both experimentally and numerically (Birch et al.,2004; Ellington et al.,1996; Liu et al.,1998; Wu and Sun,2004). This type of vortical structure, at certain high Reynolds numbers, is analogous to the spiral conical vortex generated on each leading edge of a delta wing (Birch et al.,2004). However, whereas considerable importance is attached to the force performance and power efficiency of flapping flight(Birch and Dickinson, 2003; Sun and Tang, 2002a; Sun and Tang, 2002b), few studies have addressed the detailed structure of the LEV.

In a recent flow visualization of a robotic wing moving in the motion of dragonfly hovering (Y.L., G.X.S. and W. H. Su, manuscript submitted), we observed that the LEV with intensive spanwise flow did not develop along the leading edge but moved inboard, leaving a space for the formation of a minor vortex outside the primary vortex (see Fig. 1). In fact, a similar LEV pair has been reported in butterfly free flight using smoke-wire visualization(Srygley and Thomas, 2002). This vortical system reminds us of the dual LEV structure on non-slender delta wings, which is defined as the large and small same-sense vortices (the primary vortex and minor vortex, respectively) located close to the leading edge on both sides of the secondary separation in- and outboard, respectively(Gordnier and Visbal, 2003; Taylor and Gursul, 2004). This interesting flow behavior implies that the LEV generated in flapping motions may involve sub-structures, and merit in-depth exploration.

With both visualizations the two vortices were reported to rotate in the same direction, but it is difficult to distinguish them directly from the flow images. In this paper, dye flow visualization and high-resolution DPIV measurements were conducted to reveal the detailed features of the LEV region. Based on the validation of the dual LEV, systematic investigations were performed by altering the kinematic (Re, from 160 to 3200, andα _m, from 10 to 80°) and geometric (AR, from 1.3 to 10) parameters. The well-known case of the fruit fly was re-examined to see whether the dual LEV had been missed in the previous studies. All measurements were obtained with the robotic wings flapping in the hovering condition.

The planform of the simplified model dragonfly wing (AR=5.8, AR=R/c; where R is model wing length,measured from the translational axis to the wingtip; c is mean wing chord length) served as the basis for producing the other model wings, forming a range of five AR (1.3, 3.5, 5.8, 7.5 and 10, see Fig. 2). The range of aspect ratios selected here not only covered the lowest and highest aspect ratios of insect wings (Dudley, 1990; Ellington, 1984a), but also included the case of a rotary wing. The wing planform used here was not only for simplification, but also to eliminate any geometric interference, such as the curved leading edge, from the LEV structure. Mechanical constraints meant the model wings were mounted on the tip of the rotational shaft, and the wing bases were 46 mm away from the translational axis (denoted as OT in Fig. 3A). All model wings were fabricated from 1 mm flat aluminum sheets to minimize the thickness effect(c-based normalized thickness 5%) and to allow sufficient strength and rigidity. Model wings with AR ranging from 1.3 to 7.5 mm thick all had an effective wing length (r, measured from the model wing base to the tip; see Fig. 2) of 104 mm, rendering R=150 mm. The narrowest wing (AR=10) had r=154 mm, so R=200 mm. In addition, to re-examine the case of the fruit fly wing, we used a model wing with the same planform as fruit fly Drosophila (AR=3.1, R=150, r=129).

The kinematic curves are plotted in Fig. 3B. In this study, adjusting ρ_m could achieve different values for α_m; altering T (or stroke frequency n) would yield various Re values[Re=U_tc/ν, where U_t = mean wingtip translational velocity; ν = kinematic viscosity; n = stroke frequency(Ellington, 1984b); here, since U_t=2ΦnR, Re=2ΦnRc/ν].

The flapping motions were mimicked via a self-designed electromechanical system, which contained two servo motors (Maxon Corp.,Sachseln, Switzerland) linked to a controller (Nyquist Corp., Eindhoven,Holland) controlled by the self-written software. The two motors output the translation and rotation, respectively, when a given motion was discretized and input into the personal computer. To examine the simulation quality, we sampled three sequential periods of the output motions and compared them with the ideal data. The calculated correlation coefficients (using the correlation coefficient function in Matlab 7.0.4) of translation and rotation were 0.9847 and 0.9305, respectively, validating our kinematic simulation.

Dye flow visualization offers integral pictures of the flow structures, and always serves as a reference for the DPIV measurement. In this study, dye visualization was conducted in a water tank of dimensions 600 mm×400 mm×400 mm (length×width×depth). The electromechanical system was mounted on the top of the tank. The LEVs were visualized via dyes released from a single tube (diameter roughly 1 mm), placed at the node of the leading edge and vertical to the spanwise direction so as to minimize the impact of dye releasing speed on the spanwise flow(Fig. 4). A commercial digital video (frame rate 25 frames s^-1) was used for recording. Two perspectives, i.e. viewing down from the top and horizontally along the stroke plane, were used to obtain representative flow images when the wing was traveling with different α_m values. The water was originally static, representing the hovering condition. All flow pictures were recorded after the wing had moved for more than five periods to ensure the establishment of the flow field.

DPIV measurements were also performed in the water tank, in which the water was seeded with hollow glass beads of diameter 1-5 μm. The dye-tubes on the model wings used in the dye visualizations were all removed, canceling any potential interference. A 2 mm thick laser-sheet was created by a dual-pulse Nd-Yag laser system (maximum of 200 mJ pulse^-1, LABest, Beijing,China). A frame-straddling CCD camera (1920 pixel×1080 pixel, Red Lake,San Diego, CA, USA) equipped with a lens (Micro Nikkor 105mm f/2.8, Nikon,Tokyo, Japan) was employed to achieve sufficient spatial resolution. Another CCD camera with resolution of 1008 pixel×1018 pixel (Red Lake) was used to capture the global flow field that contained both the leadingand trailing-edge vortices. The CCDs were positioned perpendicular to the light sheet. The electromechanical system was mounted on a positioning translation system (the translating direction was perpendicular to the light-sheet) on the top of the tank, and thus it was possible to measure at different spanwise locations without displacing the DPIV system. The initial positions of the wings were well adjusted, ensuring that the spanwise direction of the wing was perpendicular to the light sheet at mid-stroke. Equipped with a digital delay/pulse generator (DG 535, Stanford Research System Co., Sunnyvale, CA,USA), the laser pair was triggered and the image pair was recorded when the wing arrived exactly at the mid-downstroke position(t=0.25T). Several laserpulse separations with a range of 2-8 ms were applied to fit different Re situations. In each case,over 15 periods were sampled and averaged, excluding the first five periods(to avoid the `start-up effect'). Cross-correlation and vorticity calculations were completed using software developed by the authors. In the cross-correlation procedure, an interrogation area of 32 pixel×32 pixel with a searching step of 16 pixel was used, achieving 8100 vectors in total. Typical velocity vectors are shown in Fig. 5A. The DPIV data were post-processed through Tecplot 10. The overall DPIV arrangement is depicted in Fig. 5B.

Since only two-dimensional PIV was applied in this study, the sectional flow fields should be treated with caution for the following reasons. (1) The streamlines presented here are neither the actual stream patterns, nor their projection onto the laser-sheet plane - they should only be called sectional streamlines. However, they are helpful in providing some flow field information, e.g. the distribution of the critical points. (2) If the plane of interest is not vertical to the local vortex axis, the planar flow field topological structure may change (Dallmann et al., 1995). Thus, the absence of a focus in a vorticity-condensed region alone may be not sufficient to rule out the existence of a vortex; this case did indeed occur in our study and will be analyzed below.

Validation based on both flow visualization and DPIV, was performed as a first step to clarify the rotational senses of the two vortices. Previously,the dual LEV was observed in the downstroke when the model dragonfly wing(AR=5.8) was moving with the pattern of dragonfly hovering(non-symmetrical hovering, α_m for down- and upstrokes =60° and 30°, respectively)(Norberg, 1975). In the present study, we controlled the model wing so that it flapped symmetrically using α_m=60° for both down- and upstrokes. Φ was set to 60°, the same for the dragonfly. n was set to 0.2 Hz so that Re=1624, a value within the range of dragonfly hovering(Dudley, 2000).

Fig. 6 shows the LEV core evolution during downstroke (since the strokes were symmetrical during down-and upstrokes, we only present here the downstroke image sequence). It can be seen that two dye-concentrated lines close to the leading edge were spiraling towards the wingtip. The short one (the minor vortex) was always along the leading edge, while the inboard long one (the primary vortex) was deviating from the leading edge. This result, to certain extent, confirms our finding of the dual LEV in the prior experiment, and it also indicates that the interference of the rotational shaft and linkage (see Fig. 1), which potentially could have certain influence, evidently did not change the LEV structure. During the later half of the stroke when the wing was decelerating, the primary vortex broke down at roughly 0.35r. By contrast, the minor vortex became loose in structure (Fig. 6C). Vortex breakdown (or burst) is a dramatic change at some points of the LEV, including axial speed drop, vortex core expansion(Leibovich, 1984). The effect of the vortex breakdown is to reduce the lift of the wing.

DPIV measurement can be used to clarify the rotational senses of the two vortices and is capable of revealing the local detailed flow structure. The following DPIV measurements were all conducted at mid-downstroke(t=0.25T).

First, a section at 0.29r, where the two vortices could be distinguished easily, was studied. As shown in Fig. 7A, two vorticity-condensed regions with the same sign (negative, blue contour)located above the leeward surface near the leading edge are conspicuous. According to the spatial distribution shown in the visualization picture at 0.25T (Fig. 6B), the larger inboard vortex corresponded to the primary vortex, while the outboard counterpart was in accordance with the minor vortex. Facilitated by sectional streamlines, two foci can be identified in the centers of the two vorticity-concentrations, demonstrating the vortex nature of these two regions. As a result, the first evidence of the dual LEV structure on a flapping wing is presented. In addition, the secondary separation with opposite vorticity (yellow contour) can be detected between the two vortices just above the wing surface. These three vorticity regions constitute the basic structure of the dual LEV, reminding us of the dual-vortex structure on non-slender delta wings (Gordnier and Visbal, 2003; Taylor and Gursul, 2004). Note in Fig. 7A that a small separation layer with negative vorticity locates outboard from the minor vortex at the leading edge, where the sectional streamlines bend dramatically but form no focus. This layer can also be found in all spanwise locations in Fig. 7B. Since it is the most outboard separation structure, we name it the outboard separation.

We estimate that there are probably several reasons why the dual LEV was missed in previous experimental and numerical studies. (1) Improper tracer released location and the use of discrete tracers such as bubbles may prohibit sufficient tracers from entering and visualizing the minor vortex(Birch et al., 2004; Ellington et al., 1996). (2)The spatial resolution of CFD and PIV in prior studies may have been inadequate to reveal the fine flow structures(Liu et al., 1998; Maybury and Lehmann, 2004).(3) α_m in prior studies was in the range of 40-45°, and the PIV laser-sheets were always vertical to the spanwise direction; at this relatively low α_m the primary vortex could not be illustrated by sectional streamlines as the local vortex axis deviated from the spanwise direction (Birch et al., 2004; Dallmann et al., 1995), which will be shown below.

Further, because the flow field at the outer wing could not be visualized via this dye method, DPIV measurements were conducted along the spanwise direction from the wing-base to the tip with a 15 mm separation. It is clear in Fig. 7B that the vortex pairs are remarkable at all spanwise locations except at the wing-base. However, it should be emphasized again that the primary vortex and minor vortex in the dual LEV system refer to those located on both sides of the secondary separation in- and outboard, respectively. In this sense, the vortex pairs in Fig. 7B, which can be identified as dual LEV, are those within the range 0-0.58r. The main reason for the absence of foci in the primary vortex region at 0.43rand 0.58r is the relatively large deviation of the local vortex axis from the spanwise direction (or the laser-sheet normal), which changed the sectional topological structure (Dallmann et al., 1995). Although the primary vortex increased in size along the spanwise direction (showing a conical shape) and showed severe vorticity diffusion at the outer wing, it remained attached to the wing surface.

Questions on the foci pairs at the outer wing from 0.43rto1.00r still remain, however; what are they, and how did they form?Unfortunately, these structures could not be elucidated through current dye visualization. We estimate that the minor vortices were undergoing shedding,leaving a same-sense vortex generated behind.

In the following sections, the visualization pictures at mid-downstroke and DPIV results at 0.29r are presented.

α_m was altered from 10° to 80° with a separation of 10°. The model dragonfly wing (AR=5.8) was still used, and the same stroke amplitude and Reynolds number were retained (Φ=60°, Re=1624).

The results are shown in Fig. 8. When the wing was translating with a small α_m(below 30°), there was only a thin vorticity layer covering the leeward surface of the wing, and only a weak vortex that showed no spiral structure was created (Fig. 8A,B). Whenα _m reached 30° (Fig. 8C), two branches of concentrated LEV with considerable spanwise flows were generated. However, the LEV vorticity layer was still well connected, allowing the two vortices to interact easily with each other; such an interaction can be detected from the visualization picture in Fig. 8C as the fact that the dyes belonging to the minor vortex were being sucked by the primary vortex. The sectional streamlines fail to show the primary vortex, because the local vortex axis considerably deviated from the spanwise direction(Dallmann et al., 1995). Similar phenomena for possibly the same reason occurred whenα _m=40° (Fig. 8D). The two results may explain one of the reasons for missing dual LEV in prior studies, in which α_m was in the range 40-45° and the sampling planes such as the PIV laser-sheet were always vertical to the spanwise direction (Birch et al., 2004).

At a certain level (α_m reached 50°), both the vorticity of LEV and secondary separation were enhanced. More significantly,the LEV area was no longer connected but separated into two in- and outboard sub-regions, which were in accordance with the primary and minor vortex,respectively (Fig. 8E). Further increasing α_m (to 60° and larger), even though the two LEVs still appended above the wing surface, they were more distant from the wing. As a consequence, the two LEVs and the secondary separation all became weaker (Fig. 8F-H). The outboard separation (defined above) can also be observed in the DPIV images when α_m was sufficiently large (α_m reached 60°, see Fig. 8F-H). However, flow visualizations did not capture this structure, possibly because it was not a vortex in nature.

In sum, dual LEV existed when the wing flapped with an α_mnot less than 30°. Additionally, increasing α_m, the distance between the dual LEV and leeward wing surface was enlarged, and the two LEVs and the secondary separation all became weaker.

Again, the model dragonfly wing and flapping pattern(α_m=60°, Φ=60°) were applied,but the stroke frequency n was altered from 0.02 to 0.4 Hz to achieve a set of Re ranging from 160 to 3200.

At very low Re of 160 (roughly the Re of hovering fruit fly), there was a small crescent-like region of negative-sign vorticity near the leading edge at a considerable distance from the wing. A focus with a loose structure was in this region (Fig. 9A). The visualization picture in Fig. 9A reveals that only a very thin dye line was developing outboard to the trailing edge, while it failed to show a compact spiral structure. Compared with the case of Re=160, when Re=320, the vortical layer was closer to the wing and was stronger. Though the dye line at the primary vortex position was thicker and showed some spiral appearance, it still failed to thoroughly roll up into a concentrated vortex core (Fig. 9B). Also in Fig. 9B, the failure to represent this spiral structure viasectional streamlines could also be attributed to the larger deviation of the local vortex axis from the spanwise direction. Further increasing Re(from 640 to 3200), the negative vorticity-concentration rolled more tightly and finally divided into two individual regions, resulting in the formation of two concentrated vortices (Fig. 9C-G). Meanwhile, the secondary separation was strengthened. At the highest levels (Re=2400-3200), the primary vortex broke down at midstroke, earlier than for the lower Re cases(Fig. 9F,G). Unlike theα _m situations, the relative positions of the LEVs and the leeward wing surface did not change dramatically as Re increased. Additionally, the outboard separation became more remarkable when Rewas increased.

In general, dual LEV existed only when Re reached the level of roughly 640.

Aspect ratio (AR) is a simple but important geometric variable characterizing the shape of a wing. A range of AR from 1.3 to 10 was studied to see whether and how the geometric factor affected the dual LEV. Theα _m was set to 40° to fit the common situation in hovering insects (Ellington, 1984b).

First, clarification of the dual LEV in butterfly free flight(Srygley and Thomas, 2002) was conducted. A model wing with AR=1.3 [the same as butterfly Morpho peleides Butler (Dudley,1990)] was mounted on the flapper and moved with Φ of 150°. n was adjusted to 0.05 Hz, so that Re=4504, within the range of butterfly flapping flight(Dudley, 1991). From the visualization picture in Fig. 10, one can see a similar dual LEV pattern as discussed previously. Two same-sense vortices are also evident in the DPIV image. Therefore, dual LEV was confirmed for the low AR flapping wing, in addition to the condition with large AR.

Next, we complemented the other AR cases to gain more systematic insight into the effect of AR. As an intermediate case, AR=3.5 was conducted (Φ=150°, n=0.05 Hz, Re=1673). Again, a similar flow phenomenon was obtained. ARwas further increased to 7.5 and then to the extreme situation of 10, into the level of rotary wing. In these two experiments, Φ=60°, following the manner of the dragonfly case (AR=5.8). n was 0.25 Hz and 0.2 Hz, respectively, so Re was at the level of 1600 (Re=1561 and 1666 for the cases of AR=7.5 and 10, respectively). Surprisingly,the dual LEV structure was also evident on these very narrow wings, as with the lower AR cases. The failure to illustrate the primary vortex in the case of AR=5.8 and 7.5 via sectional streamlines could be also due to the large deviation of local vortex axis from the spanwise direction (Dallmann et al.,1995).

In general, the dual LEV system on a flapping wing was not sensitive to AR.

The famous fruit fly case (AR=3.1) was restudied. Re was adjusted to a relatively high value of roughly 1889(α_m=40°, Φ=150°, n=0.05 Hz) to allow the formation of concentrated LEV with intensive spanwise flow(Birch et al., 2004).

Again, the dual LEV was obvious (Fig. 11A,B). Considering the fruit fly wing emphasizes a curved leading edge, this experiment further indicates that the impact of the geometric factors is not as significant as the kinematic counterparts. The spanwise flow structures were also investigated (Fig. 11C), with similar results: the minor vortex at the outer wing was shed, resulting in a same-sign vortex created behind. Compared with the case of dragonfly wing with large AR (=5.8), the primary vortex on fruit fly wing diffused conspicuously only near the wingtip, showing a more stable structure.

The LEV on a flapping wing contains sub-structures that are very similar to the flow structures on non-slender delta wings(Gordnier and Visbal, 2003; Taylor and Gursul, 2004).

In the domain of delta wing, there have been several explanations to describe the formation of the dual LEV. For instance, Gordnier and Visbal stated that: “The development of the second primary vortex results from the impingement of the secondary separated flow on the primary shear layer splittingit into two same-sign vortices”(Gordnier and Visbal, 2003),which stressed the role of the secondary separation. Taylor and Gursul stated that: “With increasing Reynolds number, the significant interaction between the boundary layer and primary vortex results in this dual vortex structure” (Taylor and Gursul,2004), which emphasized the boundary-vortex interaction. Henning et al. suggested that “The topological flow structure changes on the one hand because the cone angle of the primary vortices is greater than this of a highly swept delta wing and on the other hand because the vorticity in the vortex core is not as strong”(Henning et al. 2005), which mentioned the effect of primary vortex cone angle and its relatively low vorticity.

Return to the current issue of flapping wing, neither of the above possibilities could be ruled out. Our preliminary estimation was that the secondary separation may play an active role in the minor vortex formation:while the separated boundary layer at the leading edge was rolling up to form the primary vortex, the induced secondary separation simultaneously induced the outboard separated flow to roll up into the minor vortex. However, the secondary separation alone might not be enough, because in some conditions the dual LEV was remarkable while the secondary separation was very weak. For instance, when α_m=80°, even near the wing-base the primary vortex was distant from the wing, resulting in a feeble secondary separation (Fig. 8H). Also at the outer wing, especially near the wingtip where the primary vortex diffused dramatically, a strong secondary separation had no chance to be induced(Fig. 7B). Therefore, we estimate that the negative-sign vorticity layer outboard from the minor vortex, i.e. the outboard separation, could potentially play certain role in transporting vorticity to the minor vortex, especially when the secondary separation no longer had substantial influence.

In addition, whereas the present study is based on hovering condition, the previous observation of free butterfly flight(Srygley and Thomas, 2002)suggested that the existence of the dual LEV structure may be beyond the hovering domain.

According to the above experimental facts and analyses, it can be concluded that dual LEV structure on a flapping wing can be created whenα _m and Re reach a certain level, i.e.α _m≥30°, Re≥640, implying that kinematic parameters place a certain impact on the dual LEV generation. On the other hand, dual LEV not only existed in a wide range of aspect ratios, but also on a wing with a curved leading edge, suggesting that the effect of geometric variables is relatively slight. Note that on a delta wing the dual-vortex structure exists only within a range of sweepback, angle of attack and Reynolds number (Gordnier and Visbal,2003; Taylor and Gursul,2004), the universal existence of flapping wings' dual LEV is a particularly interesting phenomenon. Probably, such a vortical system may be a basic flow structure of flapping wings.

The flow field at different spanwise locations was investigated based on DPIV for both the large AR dragonfly wing and low AR fruit fly wing. In both cases, even though unstable and diffused at the outer wing due to the reverse pressure gradient near the wingtip (the `wingtip effect'),the primary vortex remained attached on the leeward wing surface. Nevertheless, at the outer wing the minor vortex showed shedding appearance,leaving a same-sense vortex formed behind.

The complexity of the three-dimensional vortical structure shown in present study, which involves the multiple-vortex system and different flow behaviors along the spanwise direction, may be beyond our previous knowledge. Considering the technical deficiency of the employed dye method and two-dimensional DPIV measurements, in-depth study on the outer wing flow pattern is necessary.

Topological analysis has played an important role in the study of three-dimensional separated flows (Su et al., 1990). The complex nature of the flow field around a flapping wing, for example the high time-dependency, may be beyond the theoretical framework of the topological method. Nevertheless, with caution in mind, the topological analysis is still helpful in shedding some light on the skeleton of the global flow field. In this sense, according to the above results revealing the dual LEV details and a DPIV image capturing the full flow field that contained both the leading- and trailing-edge vortices (see Fig. 12A), the global topological structure of the sectional flow field is presented.

Although the flow fields in front of the windward surface were lost owing to the nontransparency of the aluminum wing, the flow details in the boundary layer were overshadowed by the laser reflection, and the critical points at far-field were out of the sampling window, they can still be deduced indirectly based on the topological rule: for a continuous vector field cut by a simply connected body, the number of nodes equals to the number of saddles(including semi-saddles on the body) minus one(Perry and Chong, 2000). The topological structure of the global sectional flow field is sketched in Fig. 12B. There are three foci(F) in total (two belong to the dual LEV and the rest to the shed trailing edge vortex), two saddles (S; one connects the dual LEV and the other is in the far-field) and four semisaddles (S′; on the wing windward and leeward surfaces). Note that topologically, node and focus are equivalent, the relation of the number of foci and saddles is 3-(2+4×1/2)=-1, obeying the topological rule.

In this experimental study, the dual LEV is confirmed for the first time on flapping wings. Insensitive to aspect ratio, such a vortical system could be created when α_m and Re reached a certain high level. Whereas the present experiments were performed in the hovering condition,considering the observation of Srygley and Thomas(Srygley and Thomas, 2002),the dual LEV may also probably exist in other free flight conditions such as forward flight. In general, dual LEV may be a basic flow structure of flapping wings.

The primary vortex did not develop along the leading edge but moved inboard. Although diffused at the outer edge, it remained attached to the leeward wing surface. On the other hand, the minor vortex formed along the spanwise direction, but shed at the outer wing, leaving a same-sense vortex created behind. Despite the existence of the sub-structures, their contributions to the general aerodynamics may be minor compared with those generated by the attached primary vortex. As far as the minor vortex formation is concerned, the secondary and outboard separations may both play active roles, and the outboard separation probably serves as a vorticity source in the shedding process of the minor vortex. Nevertheless, the present visualization and DPIV can hardly provide more information on the 3-D structure of this complex multiple-vortex system. Therefore, further visualization works that can effectively elucidate the flow situation at the outer wing, using 3-D PIV techniques such as stereoscopic DPIV (DSPIV)equipped with proper vortex identification method, and systematic study covering different phases in a stroke period, are desirable to understand the mechanism of the dual LEV formation and the global flow structure.

List of symbols
 
• AR

  aspect ratio (R/c)

 
• c

  mean wing chord length

 
• DPIV

  digital particle image velocimetry

 
• LEV

  leading edge vortex

 
• n

  stroke frequency

 
• r

  effective model wing length

 
• R

  model wing length

 
• Re

  Reynolds number

 
• t

  time

 
• T

  stroke period

 
• U_t

  mean wingtip translational velocity

 
• α

  angle of attack

 
• αm

  mid-stroke angle of attack

 
• φ

  instantaneous translational angle

 
• Φ

  stroke amplitude

 
• ρ

  instantaneous rotational angle

 
• ρm

  maximum rotational angle

 
• ν

  kinematic viscosity

We express appreciation to the anonymous referees for the insightful comments. We also thank Wen Han Su for the helpful discussion. This work is supported by Chinese Natural Scientific Foundation CNSF10472011.