3.1. Overview of lift generation

The lift generation of all cases is summarized in figure 3, with the lift coefficient ($C_L$) as the lift normalized by 0.5$\rho (\dot {\varphi }_0R_g)^{2}bc$. The first cycle is analysed here, and the cycle-averaged lift coefficient is the mean of $C_L$ over this cycle. According to a pre-validation, a comparison of the forward and backward strokes within the first cycle can mostly unfold the physics of wing rotation under a fully developed flow condition. As shown in figure 3(a), an advanced rotation ($\tau _{\alpha _0}>0$) can outperform either a symmetric rotation or a delayed rotation at almost all rotation durations ($\Delta \tau _{\alpha }$) in this research. However, even a slightly delayed rotation can significantly attenuate the cycle-averaged lift. Within our parameter space, the impact of $\Delta \tau _{\alpha }$ on the cycle-averaged lift is trivial for symmetric rotations, while a milder rotation (a larger $\Delta \tau _{\alpha }$) can reduce the lift benefit of advanced rotations and enlarge the lift loss of delayed rotations. Therefore, the lift maximum within our parameter space is achieved by a rapid advanced rotation ($\tau _{\alpha }=0.25$ to $\tau _{\alpha }=0.5$, and $\Delta \tau _{\alpha }<0.2$).

Figure 3. Analysis of lift generation: (a) cycle-averaged lift; (b) diagram of typical wing motions indicating magnitude and orientation of force vectors (red arrows); and (c) transient lift.

The wing sections of three typical rotation patterns at $\Delta \tau _{\alpha }=0.2$ are shown in figure 3(b) (with force vectors), and more detailed transient lift profiles of advanced, symmetric and delayed rotations are compared in figure 3(c). According to figure 3(c), the transient lifts of all rotation patterns have reached an identical steady state in the forward stroke before rotations ($\lambda =2\unicode{x2013} 4$), corresponding to a fully developed LEV. Also, regardless of rotation patterns, the transient lift recovers to a similar level (a steady $C_L$ close to 2) before the end of the backward stroke ($\lambda =7.5$), proving that the 4-chord-length travel is sufficient to eliminate the influence of wing rotation on the steady-state lift. For a fully delayed rotation ($\tau _{\alpha }=-0.5$), the transient lift experiences evident reduction since the onset of wing rotation ($\lambda =4$) and then recovers gradually ($\lambda =4\unicode{x2013} 6$ in figure 3c-iii). A milder delayed rotation (a larger $\Delta \tau _{\alpha }$) can enlarge this lift loss. As shown by the diagrams in figure 3(b), despite the total aerodynamic force being enhanced at the beginning of the backward stroke in the fully delayed pattern, the transient wing posture results in a downward projection, i.e. a negative lift. As the nose-down rotation proceeds, the initial peak of the total aerodynamic force is reduced and then increases to a secondary peak in the constant-speed sweeping.

For symmetric rotations ($\tau _{\alpha }=0$), as $\Delta \tau _{\alpha }$ increases, the lift benefits over the nose-up rotation period are mostly featured by a slight kink before the deceleration of the forward stroke ($\lambda =3\unicode{x2013} 4$ in figure 3c-i). Once the forward stroke enters its deceleration, the transient lift is manipulated by the change in sweep speed other than the variation of $\Delta \tau _{\alpha }$ (i.e. the reduced rotation rate). In the following backward stroke, a steeper symmetric rotation (a lower $\Delta \tau _{\alpha }$) can enhance the lift peak in the early stage of the acceleration and then result in a faster recovery towards the steady-state lift. Meanwhile, the contribution of this lift peak in the cycle-averaged lift is limited at a steeper rotation because it lasts for a shorter period (figure 3c-i). Thus the effect of $\Delta \tau _{\alpha }$ on the cycle-averaged lift of a symmetric rotation pattern is marginal. More importantly, regardless of $\Delta \tau _{\alpha }$, the lift peak in the acceleration of the backward stroke (white circle in figure 3c-i) is inferior to its counterpart in the forward stroke (black circle in figure 3c-i). Since the pitching angle ($\alpha$) is fixed in the acceleration of the forward stroke (black circle in figure 3c-i), this implies that the nose-down rotation during the start of each stroke in a symmetric pattern can postpone the LEV formation. Note that in the backward stroke, the wing also interacts with the wake vortices (including an LEV) formed in the forward stroke, and no remarkable lift benefit occurs in the symmetric rotation.

For fully advanced rotations ($\tau _{\alpha }=0.5$), the wing has just finished the rotation at the stroke reversal, thus the kinematics of forward and backward strokes are identical, except that the backward stroke involves the wing–wake interaction. As shown in figure 3(c-ii), akin to the symmetric pattern, the start of rotation first triggers a slight lift increase, followed by a continuous lift reduction until the stroke deceleration ($\lambda =2\unicode{x2013} 4$). According to the diagrams in figure 3(b), this lift reduction in the advanced pattern is mostly attributed to the anticlockwise rotation of the aerodynamic force vector, thus the enlarged aerodynamic force during this period is mostly transformed into drag. As the wing enters the backward stroke, the lift can jump up to a sufficiently higher value (white circle in figure 3c-ii) than that at the same phase in the forward stroke (black circle in figure 3c-ii). This phenomenon occurs at $\Delta \tau _{\alpha }>0.15$ and becomes prominent at approximately $\Delta \tau _{\alpha }=0.25$, indicating that the lift benefits of an advanced rotation are valid only when the wing–wake interaction occupies a sufficient period of the stroke. A higher lift peak in the acceleration of the backward stroke can further promote lift recovery in the rest stroke. However, the lift loss in the forward stroke ($\lambda =2\unicode{x2013} 4$) dominates the overall lift variation over $\Delta \tau _{\alpha }$ (figure 3c-ii). Therefore, the impact of a milder advanced rotation (a higher $\Delta \tau _{\alpha }$) and its connection with a beneficial wing–wake interaction is twofold. On the one hand, the disadvantage manifests as an earlier trigger of lift reduction that otherwise should occur later in the deceleration of the forward stroke. This lift reduction is mostly subject to a re-projection of aerodynamic force, and becomes more remarkable as the rotation duration increases. On the other hand, due to the wing–wake interaction, a longer rotation duration can lead to an enhanced lift peak in the acceleration of the backward stroke, contributing to lift enhancement. This benefit has nothing to do with the force projection and is solely related to flow physics, thus supporting the benefit of advanced rotations in flapping wings.

3.2. Flow analysis

3.2.1. Flow structure evolution

In this subsubsection, the flow pattern and vorticity dynamics of three typical rotation patterns (symmetric, fully advanced and fully delayed, $\Delta \tau _{\alpha }=0.2$) are examined to uncover the physics that dominates the impacts of the wing–wake interaction. The transient vortex structures ($Q=1$ for vortex boundaries in grey, and $Q=6$ for vortex cores in cyan) during the rotation are displayed in figure 4 and their further evolutions in the backward stroke are shown in figure 5. For a symmetric rotation ($\tau _{\alpha }=0$), the LEV in the inboard region tends to lose its conical structure as the wing pitches up in the deceleration of the forward stroke (figure 4), and, simultaneously, the unsteady vortex structures in the outboard region start to break down into sub-structures (breakdown vortex, BDV). Despite this, the entire vortex structure retains its attachment to the leeward surface until the reversal. As the wing enters the backward stroke and passes through the BDV, a tip vortex (TV) is quickly formed, together with the formation of an LEV beyond the mid-span, and they both elongate into the wake as the wing sweeps.

Figure 4. Transient vortex structures of different rotation patterns. The cyan, grey and yellow boxes indicate stroke phases. The camera orientation at $\alpha =100^\circ$–135$^\circ$ is flipped to show the other wing surface (the lighter isosurfaces). The sweep direction is labelled by a green arrow.

Figure 5. Evolution of vortex structures in the backward stroke after three typical rotations.

In contrast, when the rotation is fully advanced ($\tau _{\alpha }=0.5$), the LEV degradation and the formation of BDV during the pitch-up are weaker since the wing still sweeps forward and the feeding shear layer is less reduced. By the end of the rotation, since no backward stroke is imposed, the impingement of BDV only produces an attached horseshoe-like vortex system in the outboard region, including a TV. Moreover, in the backward sweep (figure 5), this horseshoe-like vortex system can initiate the formation of an attached LEV around the distal corner of the wing, despite the vortices along other edges all shed into the wake. As the sweep speed increases in the backward stroke, this locally attached LEV merges with the conventional LEV within the mid-span as a new conical structure. In addition, compared to the backward stroke after the symmetric rotation, the transient LEV after the fully advanced rotation is stronger at an identical time interval ahead of the steady state ($\lambda =5$ and 5.5). This is because the shed LEV induced by the BDV of the symmetric rotation fails to connect with the newly formed LEV in the backward stroke and thus postpones the overall LEV growth.

Although the lift performance of a delayed rotation discourages its application in flapping wings, its global vortex structures are also compared in figures 4 and 5. After the reversal, the BDV in the outboard region becomes further chaotic, and the wing encounters a stronger impingement with the BDV, which is promoted by both the backward stroke and a nose-down rotation. Meanwhile, the transient pitching angle ($\alpha$) is close to 90$^\circ$ when the wing enters the acceleration of the backward stroke. These two factors both enhance the feeding shear layer at the leading edge and trailing edge, thus leading to significantly stronger LEV and TEV (figure 4, $\alpha =100^\circ$). Despite the rapid growth, these two vortices quickly shed from edges (figure 4, $\alpha =120^\circ$ and 135$^\circ$), and a secondary LEV is formed along the span until the end of the backward stroke (figure 5, $\lambda =6$–$7$). Together with the transient lift diagrams reported in figure 3(b), the shedding of primary LEV and TEV explains the inferior aerodynamic force in the second half of the fully delayed rotation. Due to the undesirable impacts of delayed rotations, the following sections focus mainly on the comparison between symmetric and fully advanced rotations.

3.2.2. Vorticity dynamics analysis

In this subsubsection, a further comparison of the distinct vorticity evolution between symmetric and fully advanced rotations is provided. Sectional analysis at two representative slices ($0.25b$ and $0.5b$) within the stable LEV region is first shown in figure 6. According to the coordinate and integration procedure (§ 2.2), negative and positive circulations $\varGamma ^{(-)}$ and $\varGamma ^{(+)}$ can mostly represent the LEV intensity in the forward and backward strokes, respectively (as in the grey boxes in figures 6(a,c). Moreover, the data points in the yellow box denote the TEV intensity in the corresponding stroke.

Figure 6. Intensity of radial vorticity at two representative slices: (a) $\varGamma ^{(-)}$, (b) contours at $0.25b$, (c) $\varGamma ^{(+)}$, and (d) contours at $0.5b$. Here, S and A are the symmetric and fully advanced rotations, respectively.

It is evident that the LEV intensity starts to increase when the wing pitches up ahead of the reversal (figure 6(a), $\lambda =3\unicode{x2013} 4$, advanced rotation); otherwise, the LEV maintains the steady-state value until the deceleration, when an abrupt decrease occurs (figure 6(a), $\lambda =3\unicode{x2013} 4$, symmetric rotation). This LEV enhancement of the advanced rotation is accompanied by a simultaneous increase in TEV intensity (figure 6(c), $\lambda =3\unicode{x2013} 4$, advanced rotation), implying the generation (and then shedding) of an additional TEV to dynamically sustain the Kelvin circulation theorem. In the backward stroke, the LEV is subject to slightly faster growth when an advanced rotation is imposed. Note that in the backward stroke after both rotations, the LEV cannot reach the steady-state intensity of an impulsive-start stroke ($\lambda =0\unicode{x2013} 4$), indicating that, in general, the growth of a stable LEV on reciprocating wings is attenuated within most of a stroke due to wake effects. This agrees with Wu & Sun (Reference Wu and Sun2005), that the downwash can be detrimental to the LEV growth of insect-like flapping wings.

According to the contours of the radial vorticity component shown in figures 6(b,d), when the wing undertakes a symmetric rotation, the vorticity feeding from the shear layer is terminated during the pitch-up, and the LEV$_{f}$ in the forward stroke is convected towards the trailing edge. The LEV$_{f}$ tends to be elongated along the lower surface and then merges into the TEV$_{b}$ in the backward stroke. In contrast, the vorticity feeding from the shear layer is retained over the entire advanced rotation, therefore the LEV$_{f}$ keeps growing until the stroke reversal, together with the shedding of a stronger additional TEV$_{f}$. Almost no convection of LEV$_{f}$ is seen during an advanced rotation, and the majority of the LEV$_{f}$ is retained around the leading edge until the wing enters the backward stroke, acting as a vorticity storage for wing–wake interaction.

For the vortex evolution post rotations (figure 7), the stable LEV$_{b}$ after the advanced rotation achieves a higher intensity than that after the symmetric rotation in the first half of the backward stroke ($\lambda =5\unicode{x2013} 6$). Note that the contours after the symmetric rotation at $\lambda =4.5$ have been shown in figures 6(b,d) ($\alpha =135^\circ$). The transient LEV$_{b}$ after the symmetric rotation is stronger at $\lambda =4.5$, but it elongates and sheds from the leading edge during $\lambda =4.5\unicode{x2013} 5$, corresponding to the 3-D plots in figure 5. Moreover, for the advanced rotation, the residual negative vorticity below the lower surface moves towards the trailing edge during $\lambda =4\unicode{x2013} 5$, and boosts the formation and shedding of the TEV$_{b}$. At an identical time instance, the TEV$_{b}$ after the advanced rotation is convected further downstream (out of the window), indicating a more rapid expansion of the vortex loop (featured by LEV$_{b}$, TEV$_{b}$, TV and root vortex) and thus a higher transient lift (Sun & Wu Reference Sun and Wu2004). When the backward stroke proceeds into its second half ($\lambda =6\unicode{x2013} 8$), less difference in the LEV$_{b}$ intensity and location is observed, leading to comparable LEV$_{b}$ circulation (figure 6c) and lift generation (figure 3c).

Figure 7. Sectional plots of radial vorticity in the backward stroke: (a) symmetric rotation and (b) fully advanced rotation.

The critical radial vorticity transport terms are then examined to explain two unique features in the LEV evolution of the advanced rotation (figure 8), i.e. the absence of convection towards the trailing edge during the rotation, and the rapid growth of TEV$_{b}$ and LEV$_{b}$. As the wing pitches up towards 90$^\circ$, downward convection of the radial vorticity ($C_{ry}^{(+)}$, positive $C_{ry}$ to negative $C_{ry}$) occurs in the stable LEV region, which is almost perpendicular to the stroke plane and drives the movement of LEV$_{f}$ towards the trailing edge (figure 8a). Here, $C_{ry}^{(+)}$ is located within the entire stable LEV region for the symmetric rotation, while that of the advanced rotation is limited within a smaller region in the chordwise direction, and another negative $C_{ry}$ occurs along the leading edge to terminate sustained $C_{ry}^{(+)}$.

Figure 8. Critical radial vorticity transport terms within/post typical rotations: (a) vertical convection, (b) tangential convection, and (c) vortex stretching. The isosurfaces are outlined by 6 (red) and $-$6 (blue).

Moreover, the merge of the negative radial vorticity into TEV$_{b}$ during the advanced rotation can be demonstrated by the tangential convection of the radial vorticity component (figure 8b), i.e. $C_{rt}^{(+)}{\rightarrow }C_{rt}^{(-)}$ for the downstream convection of negative vorticity component in TEV$_{b}$, and $C_{rt}^{(-)}{\rightarrow }C_{rt}^{(+)}$ for the downstream convection of positive vorticity component in LEV$_{b}$. At $\lambda =4.5$, the downstream convection of TEV$_{b}$ is prominent and propagates into the wake during the advanced rotation, which is comparable to that at $\lambda =5$ for the symmetric rotation. Thus it is suggested that the residual negative vorticity component generated in the forward stroke can promote this downstream convection and thus shift the TEV$_{b}$ further away from the wing. Meanwhile, the downstream convection of LEV$_{b}$ during the symmetric rotation experiences a less regular variation over $\lambda =4\unicode{x2013} 5$ because a conical $C_{rt}^{(-)}{\rightarrow }C_{rt}^{(+)}$ is first established at $\lambda =4.5$ and then levels off along the span at $\lambda =5$. However, the downstream convection of LEV$_{b}$ during the advanced rotation is consistently enhanced in a conical form along the span since $\lambda =4$, representing a sustained and regular vorticity feeding from the shear layer, and thus a more rapid growth of LEV$_{b}$.

In addition, the vortex stretching ($S_{r}^{(+)}$) within the LEV$_{b}$ after the symmetric rotation also has remarkable transient variations during $\lambda =4\unicode{x2013} 5$ (figure 8c), in that an $S_{r}^{(+)}$ value has been established at $\lambda =4.5$ and then disappears at $\lambda =5$. Another conical $S_{r}^{(+)}$ is formed as the wing sweeps forward to $\lambda =5.5$. This re-establishment of $S_{r}^{(+)}$ is not observed in the advanced rotation, and a single conical $S_{r}^{(+)}$ is maintained and enhanced gradually until $\lambda =6$.

3.3. Coexistence of dual mechanisms

In § 3.2, the coexistence of an attached LEV (within the mid-span) and a BDV system (near the tip) has been evidenced. The downward convection of the attached LEV underneath the upwind surface (figures 6 and 7) occurs in both symmetric and fully advanced rotations, suggesting the existence of the single-vortex suction mechanism. However, the impacts of BDV on local pressure and the underlying mechanism are unclear. To further explain this span dependency of wing–wake interaction, the pressure distributions on the leeward surfaces of both symmetric and fully advanced rotations are compared in figure 9. Note that a switch of the leeward surface occurs when the wing enters the backward stroke (indicated by red boxes). Before the rotations (figure 9a), the stable LEV$_{f}$ can induce a suction region beneath its conical structure. This suction region experiences a continuous attenuation in the symmetric rotation, while that in the advanced rotation first becomes stronger before $\alpha =90^\circ$, and then shrinks as the rotation proceeds. This transient enhancement of the suction region coincides with the sustained LEV$_{f}$ growth (figures 6a,c). As the sweeping decelerates ($\alpha >90^\circ$), the stronger LEV$_{f}$ in the advanced rotation can still induce a slight suction region near the leading edge, which is explained by Lua et al. (Reference Lua, Lim and Yeo2011) as the single-vortex suction mechanism. In contrast, this phenomenon is trivial in the symmetric pattern due to a remarkably weaker LEV$_{f}$.

Figure 9. Transient pressure distributions on the leeward surface: (a) within rotations, and (b) post rotations. The switch of the leeward surface in (b) is indicated by red boxes.

Another phenomenon observed in both rotations is the local high pressure around the wingtip (figure 9a), which is mostly attributed to the BDV impingement towards the upwind surface. Both the stroke deceleration and the nose-down rotation can promote this impingement, thus it becomes more prominent for the symmetric rotation due to the overlapping of these two kinetic features. Unlike the interaction with a single vortex (i.e. the stable LEV$_{f}$) explained above, the BDV manifests as a complex vortex system with strong unsteadiness. Although it is laborious to spot detailed elements of this unsteady vortex system, its overall impact is akin to a ‘jet’ towards the wing, thus resembling the inter-vortex jet mechanism. The difference is that the mechanism proposed by Dickinson (Reference Dickinson1994) was to explain the interaction with a counter-rotating vortex pair, which can induce an inter-vortex jet towards the surface, thus enlarging the incoming velocity and stagnant pressure. The BDV is certainly different from the counter-rotating vortex pair, but the significant breakdown can slow down the swirling of the vortex system and attenuate the low pressure in the vortex core. Thus, compared to the impingement of a stable LEV$_{f}$ (the low pressure inside the vortex is retained), the BDV impingement leads to an exactly opposite pressure variation. This phenomenon is named the breakdown-vortex jet mechanism in this work. Therefore, it is suggested that the wing–wake interaction of a flapping wing at ${\textit {Re}}>10^3$ can be explained mostly by the single-vortex suction and breakdown-vortex jet mechanisms. These two mechanisms may exist simultaneously (but at different spanwise locations) during the wing rotation, especially for an advanced one.

The difference in BDV impingement between the symmetric and advanced rotations also alters the wingtip suction region in the backward stroke (figure 9b). For the fully advanced rotation, the wing encounters the BDV at $\alpha =45^\circ$ without a rotational speed, and the LEV$_{b}$ near the tip can mostly retain its attachment in the backward stroke (figure 5). Therefore, the suction region near the wingtip is prominent after the advanced rotation at $\lambda =4.5$, and it is then connected with the suction region induced by the inboard LEV$_{b}$. However, due to the stronger BDV impingement in the symmetric rotation (a higher $\alpha$ and a superposition of nose-down rotation), the LEV$_{b}$ formed by the BDV impingement quickly sheds into the wake during $\lambda =4.5\unicode{x2013} 5$, and the inboard suction region is then established until $\lambda =5\unicode{x2013} 5.5$. These explanations relate the superb aerodynamic force during the acceleration of the backward stroke in the advanced rotation (shown in figure 3b) to the stronger wingtip suction led by the locally attached LEV (which will be discussed in § 3.4).

The coexistence of single-vortex suction and breakdown-vortex jet mechanisms above the dorsal surface of the wing is further evidenced by the sectional pressure force (figure 10). For the symmetric rotation (figure 10a), the low-pressure region induced by the conical LEV on the dorsal surface becomes stronger towards the tip ($\alpha =45^\circ$, figure 10a). As the rotation starts, the LEV intensity and induced low pressure are both attenuated ($\alpha =60^\circ$–$90^\circ$, figure 10a), but a local pressure reduction still exists within the 0.67 span (#1 region), inferring the suction led by the remaining LEV. However, the dorsal pressure within the 0.33 span from the wing tip (#2 region) is remarkably enhanced, corresponding to the jet of BDV impingement. At $\alpha =90^\circ$–$135^\circ$, most lift production originates from the jet mechanism in the #2 region, whereas the suction mechanism in the #1 region dictates the lift generation in the first half of the rotation. Therefore, the single-vortex suction and breakdown-vortex jet mechanisms can coexist on a flapping wing during the rotation, but they are located within the stable and unsteady LEV regions, respectively.

Figure 10. Sectional pressure force during the rotation: (a) symmetric, $\Delta \tau _{\alpha }=0.2$, and (b) fully advanced, $\Delta \tau _{\alpha }=0.2$. The blue and red lines represent the leeward and upwind surfaces in the forward stroke, respectively, and the green mask indicates the net force. Here, #1 and #2 refer to the single-vortex suction and breakdown-vortex jet mechanisms, respectively.

The fully advanced rotation case also experiences the coexistence of dual mechanisms during rotation (figure 10b). As less LEV attenuation occurs in the first half of the rotation ($\alpha =45^\circ$–$90^\circ$), the low-pressure region caused by the suction mechanism (#1 region) is further enhanced. In contrast, the stable high pressure led by the jet of BDV impingement (#2 region, $\alpha =135^\circ$) is close to that of the symmetric rotation. The only difference related to this high pressure is that the impingement is postponed in the fully advanced rotation. Moreover, despite the fact that the rotation pattern may affect the intensity and phase (with respect to the rotation) of the two mechanisms, their valid ranges along the span are barely changed. This is because these mechanisms are dependent on the LEV status, which is dominated by the local revolving radius (Chen et al. Reference Chen, Zhou, Werner, Cheng and Wu2023b).

3.4. Lift peak due to breakdown-vortex impingement

In this subsection, we further focus on the transient lift peak in the acceleration of the backward stroke after fully advanced rotations (see the white circle in figure 3(c), $\tau _{\alpha _0}=0.5$), and compare it to its counterpart in the forward stroke (without wing–wake interaction). Since this lift peak occurs under an appropriate rotation duration ($\Delta \tau _{\alpha }>0.2$), the impact of $\Delta \tau _{\alpha }$ is also considered here, taking $\Delta \tau _{\alpha }=0.1$ and 0.3 as examples. The transient 3-D vortex structures and pressure distributions during $\lambda =4\unicode{x2013} 4.5$ (backward stroke) and $\lambda =0\unicode{x2013} 0.5$ (forward stroke) are compared in figure 11. During $\lambda =0\unicode{x2013} 0.5$ (figure 11a), the typical vortex structures are in a perfect ring shape along the edges, and the conical LEV$_{f}$ almost covers the entire span. However, in the backward stroke, regardless of $\Delta \tau _{\alpha }$ (figures 11b,c), the LEV$_{b}$ cannot achieve a comparable conical structure along the span but mostly concentrates near the wingtip at $\lambda =4.25$. This is attributed to the BDV impingement, the majority of which is located near the distal end of the wing. Comparing the BDV structures at the beginning of the backward stroke ($\lambda =4$ in figures 11b,c), they are stronger and more coherent at $\Delta \tau _{\alpha }=0.3$. This implies that a milder advanced rotation ($\Delta \tau _{\alpha }=0.3$) can alleviate the dissipation and breakdown of this complex vortex structure. These stronger wake vortices further induce an intensified interaction with the wing, leading to a stronger LEV$_{b}$ and TV$_{b}$ system ($\lambda =4.25$).

Figure 11. Vortex structure and pressure distribution with/without wing–wake interaction in a fully advanced rotation: (a) vortex structure without interaction, (b) vortex structure with interaction at $\Delta \tau _{\alpha }=0.1$, (c) vortex structure with interaction at $\Delta \tau _{\alpha }=0.3$, and (d) pressure distributions. The blue and red boxes represent different camera views, and the green arrow denotes the sweep direction.

As shown by the transient pressure at $\lambda =0.25$ (figure 11d), corresponding to the lift peak without BDV impingement, an LEV$_{f}$-induced suction region is observed along the leading edge. Together with the gradual increment of high pressure (on the upwind surface) along the span, the lift peak at $\lambda =0.25$ originates mostly from the leading-edge region. In contrast, due to the BDV impingement in the backward stroke, the high-lift region is further confined near the wingtip, and extends along the chord. The reasons for these variations are twofold. First, due to the LEV$_{b}$ concentration near the wingtip, the suction force on the leeward surface is confined near the intersection of the leading edge and wingtip. Second, due to the impingement of BDV in the wake, the high pressure on the upwind surface experiences a more prominent increase in the outboard region of the wing. As the advanced rotation becomes milder and thus the LEV$_{b}$ and BDV are stronger, these two phenomena are more remarkable. Therefore, the transient lift peak due to BDV impingement in a fully advanced rotation (during the acceleration of the backward stroke) becomes intensified as $\Delta \tau _{\alpha }$ increases to 0.3 (figure 3c).

A closer examination of the wing-tip vortex structures and the flow patterns with/without BDV impingement is shown in figure 12. Comparing figures 12(a) and 12(b), regardless of $\Delta \tau _{\alpha }$, the BDV impingement can intensify the local radial vorticity inside the LEV between $0.8b$ and $0.95b$. Moreover, the LEV$_{b}$ at $\Delta \tau _{\alpha }=0.3$ is slightly stronger between $0.8b$ and $0.95b$, and further approaches the leeward surface. In contrast, the LEV$_{b}$ at $\Delta \tau _{\alpha }=0.1$ is lifted from the surface at $0.95b$, which can be explained by additional upwash induced by the wake vortex pair above the LEV$_{b}$ (see the $0.95b$ slice in figure 12b). Usually, a strong downwash is established in the wake of flapping wings (e.g. $0.8b$ slice in figure 12b). The local upwash can mediate the existing downwash, and thus leads to a further tangential elongation of the LEV$_{b}$, and induces a lower suction force on the leeward surface (figure 11d). A similar but weaker upwash is also observed on the $0.8b$ slice at $\Delta \tau _{\alpha }=0.3$ due to the pass-by of a vortex pair. In addition, there is another obvious interaction between the upwind surface and the residual BDV. According to the flow field outlined by red boxes in figure 12(b), the approaching of this residual BDV reverses the downwash ahead of the wing to a comparable upwash, which triggers the breakdown-vortex jet that causes the local high-pressure region on the upwind surface.

Figure 12. Radial vorticity and flow vectors near the wingtip during an advanced rotation: (a) $\lambda =0.25$ (without breakdown-vortex impingement), and (b) $\lambda =4.25$ (with breakdown-vortex impingement).

3.5. Extension to previous works and limitation

Previous 3-D studies on the wing–wake interaction mostly focus on fruitfly wings at ${\textit {Re}}$ close to 100 (Sun & Tang Reference Sun and Tang2002; Birch & Dickinson Reference Birch and Dickinson2003; Wu & Sun Reference Wu and Sun2005; Kweon & Choi Reference Kweon and Choi2010; Lee & Lua Reference Lee and Lua2018). Compared to the present work, almost no LEV breakdown occurs in the outboard region at ${\textit {Re}}\sim 10^2$. This unsteady LEV feature at ${\textit {Re}}>10^3$ may introduce a locally novel wing–wake interaction phenomenon during the wing rotation, which is one of the motivations of this work. Our force analysis found that an advanced rotation enhances the overall lift of a flapping wing with these unsteady LEV features at ${\textit {Re}}$ up to over $10^3$. Similarly, delayed rotations are still detrimental to lift generation at ${\textit {Re}}>10^3$.

Upon the inter-vortex jet mechanism (Dickinson Reference Dickinson1994), despite the unsteady mix-up of LEV and TV in the outboard region, there is no strong evidence for the pair-up of LEV and TEV. Thus this mechanism may not be appropriate to explain the remarkably high pressure on the upwind surface. However, the residual breakdown vortex in the outboard region is strong and can also impinge towards the upwind surface, resulting in a phenomenon similar to the inter-vortex jet (figure 12b). In general, the impingement of the breakdown vortex always leads to a local high pressure for both symmetric and advanced rotations (figure 9a). This mechanism is novel for flapping wings at ${\textit {Re}}$ up to over $10^3$ and is thus named as the breakdown-vortex jet mechanism. In contrast, the interaction between the stable LEV in the inboard region and the wing leads to a reduced pressure (figure 9a), equivalent to the single-vortex suction mechanism proposed by Lua et al. (Reference Lua, Lim and Yeo2011). These findings support that the wing–wake interaction during the rapid rotation of 3-D flapping wings at ${\textit {Re}}>10^3$ is dependent on the local LEV state. More importantly, the breakdown-vortex jet (beneficial) and single-vortex suction (detrimental) mechanisms may occur simultaneously but within different spanwise regions.

In the 3-D simulations done by Lee & Lua (Reference Lee and Lua2018), the wing–wake interaction at ${\textit {Re}}=10^2$ is explained by four mechanisms, among which the LEV shedding due to vortex pairing (detrimental) and the formation of a closely attached LEV (beneficial) are unique for a 3-D flapping wing. In our research, the first mechanism is absent in both symmetric and advanced rotations since the stable LEV generated in the inboard region of the wing is convected towards the trailing edge and then merges with the newly formed TEV (figures 6b,d), instead of pairing up with the new LEV near the leading edge. The second mechanism implies that the wake can improve the LEV attachment by inducing a downwash, and thus increases the lift generation after the mid-stroke. Apart from the explanation proposed by Lee & Lua (Reference Lee and Lua2018), the downwash effect around the mid-stroke is mostly regarded to be adverse to lift generation (Sun & Tang Reference Sun and Tang2002; Oh et al. Reference Oh, Lee, Park, Choi and Kim2020; Cai et al. Reference Cai, Kolomenskiy, Nakata and Liu2021). This is because the downwash can reduce the effective angle of attack of the wing. Although the interactive effects of wings and downwash around the mid-stroke are not the focus of this work, our data show that the lift profiles after the mid-stroke of both forward and backward strokes are almost identical for symmetric rotations (figure 3c). This observation is retained in the following ten flapping cycles (lift profiles over multiple flapping cycles are provided in the Appendix). Together with the contradictory findings of Lee & Lua (Reference Lee and Lua2018), Sun & Tang (Reference Sun and Tang2002), Oh et al. (Reference Oh, Lee, Park, Choi and Kim2020) and Cai et al. (Reference Cai, Kolomenskiy, Nakata and Liu2021), our results indicate that the specific contribution of the wing–downwash interaction to lift generation may be sensitive to wing kinematics.

Although a slight lift enhancement occurs over the same period after an advanced rotation, it is suggested that due to the distinct LEV states in the inboard and outboard regions, the local wing–wake interaction may involve a remarkable spatial difference in the specific mechanism. Evidence for this is the upwash, rather than a downwash, existing above the wing-tip LEV after an advanced rotation (figure 12b). This upwash is induced by a vortex pair originating from the breakdown vortex generated in the previous stroke. Due to the strong unsteadiness of this vortex structure and its dependency on wing kinematics, the location of this upwash along the span is closely associated with $\Delta \tau _{\alpha }$. Again, once this upwash replaces the downwash, the fourth mechanism proposed in Lee & Lua (Reference Lee and Lua2018) is locally invalid since the LEV$_{b}$ is lifted from the surface and the suction force is thus reduced.

As suggested by Wu & Sun (Reference Wu and Sun2005) and Lee & Lua (Reference Lee and Lua2018), the contribution of wing–wake interaction on the lift generation of 3-D flapping wings is strongly related to wing kinematics and may even be a case-by-case problem. Within this research, the effects of rotation duration and phase relative to the stroke are examined, while no discussion on sweep kinematics is involved. Future studies can conduct a coupled variation of sweep and rotation kinematics, and further justify our findings in a broader parameter space. Moreover, our previous works on the LEV dynamics of flapping wings have quantified the connection between local LEV status with wing kinematics covering a wide space over ${\textit {Re}}$, AR, and Ro (Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022a, Reference Chen, Zhou, Werner, Cheng and Wu2023b). An extension for the quantitative connection between local LEV status and the mechanisms of wing–wake interaction is encouraged. Due to the strong unsteadiness in the wing–wake interaction, an explicit model can be impractical, and black-box models using machine learning, e.g. deep neural networks, should be more efficient.