3.1 Aerodynamic performance

The forces on the insect’s body and wings were computed through direct integration of the surface pressure and shear. The lift force ( $F_{L}$ , along the vertical direction) are presented here as non-dimensional coefficients, which were computed by, $C_{L}=F_{L}/0.5{\it\rho}U_{tip}^{2}A_{wing}$ , where $C_{L}$ is the lift coefficient. Here $U_{tip}$ is the mean wing tip velocity ( $U_{tip}=3.1U_{f}$ ), which is defined as $U_{tip}=(1/T)\int _{0}^{T}\sqrt{u_{tip}^{2}+v_{tip}^{2}+w_{tip}^{2}}\,\text{d}t$ , where $u_{tip}$ , $v_{tip}$ and $w_{tip}$ are the tip velocity components in $x$ , $y$ and $z$ directions, respectively. We use $A_{wing}$ to denote the area of the wing.

Here, the comparisons of the instantaneous forces on the cicada body and the right wing are shown in figures 5(a) and 5(b), respectively. In each plot, solid lines indicate simulation results from the WB case while dashed lines represent either the BD or WN model. Cycle-averaged drag shows that the wing–body system in the WB model experiences a very small drag force (less than 2.9 % of the overall lift force). This is consistent with our experimental observation, in which the cicada was performing a slightly decelerating forward flight. The cycle-averaged lift and their relative enhancement ( ${\rm\Delta}\bar{C}_{L}=(\bar{C}_{L}|_{WB}-\bar{C}_{L}|_{WN/BD})/\bar{C}_{L}|_{WN/BD}$ ) due to WBI are shown in table 2. According to figure 5(a), most of the wing lift is generated by the downstroke, which is in line with the previous study (Yu & Sun Reference Yu and Sun2009). The average lift enhancement on wings due to the WBI is about 6.7 %. This is slightly higher than that found in previous studies (Aono et al. Reference Aono, Liang and Liu2008; Yu & Sun Reference Yu and Sun2009; Liang & Sun Reference Liang and Sun2013), which is due to the stronger WBI.

Figure 5.

Time traces of lift and thrust coefficients of the body (a) and a single wing (b) during the fourth flapping cycle when both the aerodynamic forces and flow reach a periodic state. Peak 1 and 2 in figure (a) occur at $t/T=0.3$ and 0.74, respectively.

Table 2.

Cycle-averaged lift coefficient and its enhancement due to WBI.

The overall lift force produced by the wings and the body together is increased by about 18.7 % according to table 2, which suggests a significant aerodynamic benefit generated by WBI. This is mainly because of the lift increment on the cicada body ( $0.109-0.013=0.096$ ), which contributes about 65 % of the total lift enhancement ( $0.930-0.791=0.148$ ). The body lift accounts for about 11.6 % of the total lift generation in the WB model, whereas the body can only produce less than 2 % of the total lift when separated from the wings and simulated under the same flow conditions. Therefore, the body plays a more important role in force generation when it interacts with the flapping wings. A clear correlation between the instantaneous forces produced by the body and the periodic flapping motion of the wings can be observed in figure 5. The WB produces periodic forces with much larger amplitudes than that of BD model. In addition, the peak force values generated by the body for the WB model occur midway through the downstroke, which follows the force peaks generated by the wings. This further indicates that the lift enhancement on the body is associated with WBI.

Figure 6.

A series of instantaneous wake structures, visualized by ${\it\lambda}=2.5$ , during downstroke for WB model (a–c) and WN model (d–f): (a,d) $t/T=0.14$ ; (b,e) $t/T=0.3$ ; (c,f) $t/T=0.48$ .

3.2 Wake topology

The 3D vortex structures are visualized by the isosurface of the imaginary part of the complex eigenvalue ( ${\it\lambda}$ ) derived from the instantaneous velocity gradient tensor, which identifies flow regions where rotation dominates over strain (Mittal & Balachandar Reference Mittal and Balachandar1995). Since the BD model generates very little lift force compared with the other two models, this section mainly focuses on the flow features due to WBI by comparing the vortex structures in WB with those in WN.

Figures 6 and 7 show the comparison of instantaneous wake structures of the WB model and the WN model during the downstroke and upstroke, respectively. The WB model is on the top row (labelled as a–c, see also supplementary movie 1 available at http://dx.doi.org/10.1017/jfm.2016.175) and the WN model is on the bottom row (d–f, see also supplementary movie 2). In general, the vortical structures including the LEVs and the tip vortices around the wings are identical in both cases. This is consistent with those found in the previous studies of insect forward flight (Yu & Sun Reference Yu and Sun2009; Liang & Sun Reference Liang and Sun2013). However, there are two unique sets of vortical structures generated by the cicada body, the thorax vortex (TXV) and the posterior body vortex (PBV), which were never reported in previous studies on other insects. Specifically, the TXVs are developed from the upper surface of the body thorax and then shed onto the wings. The PBVs are spiraled vortex tubes, which emanate from the posterior region of the body and entangle with the RVs. As a result, the RVs in the wing root region (labeled V1 in figures 6 and 7) are formed very differently in these two cases. From figure 6, in the downstroke, the RVs in WB model are entangled with the PBVs. However, in WN, the RVs are found developed completely with stronger profiles (see figure 6 b,c,e,f). During the upstroke (figure 7), the RVs in WB model are shed from the wings at the early stage and then merged with the downstream wakes, while in WN model they stay attached to the wing root until the end of the stroke. Results have also shown that the strengths of the body vortices are comparable with those of wing vortices (figure 6 b,c). This indicates the potential aerodynamic contributions from the body vortices need to be evaluated separately. Next, we will describe the formation and the development of the two kinds of body vortices and their aerodynamic contributions.

Figure 7.

A series of instantaneous wake structures, visualized by ${\it\lambda}=2.5$ , during upstroke for WB model (a–c) and WN model (d–f): (a,d) $t/T=0.74$ ; (b,e) $t/T=0.9$ ; (c,f) $t/T=0.98$ .

In the early stage of the downstroke (figure 6 a), when the wing is still close to the body, only a pair of PBVs are found connecting the body and the wings of the cicada. There is no visible TXVs identified by the 3D isosurface with a value of ${\it\lambda}=2.5$ . As the wings flap down, the TXVs are then generated such that distinct vortex tubes starting from the thorax and shedding onto the wings can be clearly observed at the mid-downstroke (figure 6 b). Meanwhile, the PBVs grow stronger and tangle with the vortex tubes shed from the wing roots. As the wing flaps down continuously, one side of the PBV detaches with the wing surface while the other side remains attached to the body (figure 6 c). However, the TXVs are always attached to the body and keep growing until the end of the downstroke before wing reversal. After the reversal, the PBVs will detach from the body and convect downstream. The TXVs merge with LEVs as the wings are squeezing the flow during upstroke. At this point, visible frontiers between TXVs and LEVs cannot be found any more (figure 7 a). As the wings further flaps upward, the TXVs slightly move backward along the body but are never shed (see both figure 7 b,c). Interestingly, the TXVs at the end of the upstroke become the rudiments of the PBVs for the following downstroke (figure 7 c).

According to both the aerodynamic forces and vortex dynamics discussed above, there is a clear correlation between the force behaviour and the body-generated vortices in the cicada flight. The first peak of the body lift shown in figure 5(a) is at about $t/T=0.3$ when both the TXVs and the PBVs are well developed. After $t/T=0.3$ , the PBVs are prone to shed and the body lift is about to decrease. The second lift peak occurs at about the time that the TXVs and LEVs are compressed by the upward flapping of the wing. As a result, the profiles of TXVs are found even larger when compared with the downstroke. These correlations imply that the lift force enhancement on the body is related to the formation and evolution of TXVs and PBVs. This will be further discussed in § 3.3.

In order to further understand the aerodynamic importance of the TXVs and PBVs, we compare spanwise vorticity ( ${\it\omega}_{z}$ ) contours of the slice-cuts close to the wing root region between the WB and WN models (figure 8). In WB, two slice-cuts along the body (slice 1) and the wing-chord (slice 2) are shown at the mid-downstroke and mid-upstroke in figures 8(a) and 8(c), respectively, whereas in figure 8(b,d) only the corresponding slice-cut along the wing-chord are shown due to the non-existence of the body. It can be seen in the slice 1, a strong vortex with the same orientation as the wing LEV is formed and attached to the body thorax and another vortex can be seen above the posterior body part. These two vortices correspond to the TXV and the PBV shown in figures 6 and 7. As such, the wing LEVs in slice 2 (figure 8 a,c) are significantly different from those in WN model (figure 8 b,d). They are greatly strengthened due to the involvement of the TXVs and the PBVs. This indicates a considerable amount of force increments on both the body and the wing root region.

Figure 8.

Slices of the vorticity contour ( ${\it\omega}_{z}$ ) in the near wake for WB model (a,c) and WN model (b,d): (a,b) $t/T=0.3$ ; (b,d) $t/T=0.74$ . In WB, slice 1 is taken along the body at $0.17L$ away from body middle plane and slice 2 is cut along the wing chord at $0.1L$ away from the wing root. In WN, the position of the slice is the same as that of slice 2 in WB. The vorticity is normalized by $U_{f}/c$ .

To further illustrate the reinforcement of the LEVs near the wing root region due to the body vortices, we plot a series of spanwise vorticity ( ${\it\omega}_{z}$ ) contours along the wing span for both WB and WN models at the mid-downstroke in figure 9(a) and (b), respectively. At this time instant, the wings are nearly parallel to $z$ -axis, so the contours of ${\it\omega}_{z}$ actually represent the shape of LEVs on the wings. It is found that the LEVs in both models are almost identical to one another in the area away from the wing root ( ${>}0.7L$ ), whereas substantial difference of wing LEVs can be observed near the wing root. Specifically, multiple vortex cores have been generated by TXVs and PBVs (e.g. at $0.1L$ and $0.3L$ ) for the wing LEVs in the WB model. As a result, the overall strength of the wing LEVs have been enhanced by WBI. Figure 9(c) shows the mean circulation of the spanwise vorticity during downstroke for both WB and WN models as well as the circulation difference between these two models ( ${\it\Gamma}_{WB}-{\it\Gamma}_{WN}$ ). We found that the circulation of the spanwise vorticity over the wing root region is significantly enhanced, for example, at $0.1L$ , it is increased by 78 %.

Figure 9.

Comparison of spanwise vorticity ( ${\it\omega}_{z}$ ) contour between (a) WB model and (b) WN model. The slices are taken from the wing root to the wing tip and the vorticity is normalized by $U_{f}/c$ . (c) Comparison of the averaged circulation of ${\it\omega}_{z}$ over downstroke between the WB model and WN model. Here $z_{s}$ represents the position along the wing span.

3.3 Surface pressure

In this section, pressure distribution on the body and the wings are analysed. The pressure coefficient is defined as $C_{p}=(p-p_{\infty })/0.5{\it\rho}U_{tip}^{2}$ , where $p_{\infty }$ is the pressure in the freestream. The above force comparison indicates that the lift on the body increased a lot, so we plot the surface pressure on the body in figure 10. Figure 10(a) is the pressure on the body in the WB model at $t/T=0.3$ at which the first peak of the body lift is observed (see figure 5 a). Two low-pressure regions on the thorax and posterior body can be clearly observed at this moment, which coincide with the locations of TXVs and PBVs, implying that they were generated by the two vortices. At the time instant $t/T=0.74$ , at which the second lift peak occurs (see figure 5 a), the low-pressure regions generated by the PBVs disappears (figure 10 b), leaving only one low-pressure region on the body. This is because PBVs have already detached with the body. In addition, during upstroke, this low-pressure region moves to the top surface of the body and follows the movement of TXVs (figure 10 c). According to this analysis about the spacial and temporal change of the pressure on the body, the high body lift generated during downstroke is attributed to both TXVs and PBVs, while the lift peak which occurs at the mid-upstroke is mainly due to the TXVs.

Figure 10.

Surface pressure coefficient distribution on the body for the WB model at selected instants: (a) $t/T=0.3$ ; (b) $t/T=0.74$ ; (c) $t/T=0.9$ .

To compare the surface pressure on the wing, we plot the difference of the pressure coefficient between the lower surface and the upper surface ( ${\rm\Delta}C_{p}$ ) in figure 11. Figure 11(a,b) indicate that the overall features of the pressure on the wing in WB and WN are similar. The outer half of the wing, especially on the leading-edge region, generates most of the lift force. The difference of ${\rm\Delta}C_{p}$ between WB and WN is mainly located on the wing root region (see figure 11 c). An average 6.5 % enhancement in wing lift production can be found in the WB model. This is attributed to the aforementioned enhancement of the strength of the LEVs near the wing root.

Figure 11.

Pressure difference between the upper and lower wing surface ( ${\rm\Delta}C_{p}$ ) of the right wing for the WB model (a) and WN model (b), respectively. (c) Shows the difference of ${\rm\Delta}C_{p}$ between (a) and (b).

Isosurface contours of a low-pressure level have been plotted in figure 12(a) and (b) for WN and WB, respectively, at $t/T=0.3$ . These figures represent the same mid-downstroke instant of the wing as shown in figure 6(e) for the WN model and in figure 6(b) for the WB model, respectively. Noticeable in both plots is a large region of low pressure right behind the wings which is due to the LEVs, RVs, forewing tip vortices (FTVs) and hindwing tip vortices (HTVs). However, for the WB model, an additional low-pressure region can be found behind the head and the thorax, which extends to the region between the sides of the abdomen and the hindwings. Figure 12(c) and (d) show the schematic of the correlation between vortex structures and regions of low pressure in figure 12(a) and (b), respectively. The major vortical structures of the WN and WB models are identified. According to figure 12, the extension of the low-pressure region is due to the distinct and strong vortex structures, TXVs and PBVs, created by WBI. As a result, lift production on both the body and the wings were enhanced.

Figure 12.

Isosurface of pressure coefficient ( $C_{p}=-0.2$ ) at $t/T=0.3$ for (a) the WN model and (b) the WB model. Wake schematic for (c) the WN model and (d) the WB model at $t/T=0.3$ .

3.4 Effects of key parameters on lift enhancement due to WBIs

In this section, we examine the effect of three key parameters which are important to wing–body, body and wings, respectively, on the lift enhancement between the WB models and WN models. These parameters are the minimum wing–body distances ( ${\it\delta}$ ), body inclination angles ( ${\it\chi}$ ) and reduced frequency ( $k$ ). For a range of each parameter, the lift forces computed in the WB models are compared with the results from companion WN and BD models in order to assess the difference of the corresponding vortex dynamics and aerodynamic performance. Only major conclusive results are listed below.

First, we focus on the effect of the minimum wing–body distance. For this analysis, the reduced frequency and the body inclination angle are fixed at 0.83 and $28^{\circ }$ , respectively. Figure 13(a) shows the variation in the cycle-averaged lift coefficients with the minimum wing–body distance for WB and WN. It is found that for all cases, the lift decreases monotonically with ${\it\delta}$ . The corresponding relative increase in the lift with the minimum distance is shown in figure 13(b). The plots show that for a given flapping flight, the gain in lift for ${\it\delta}<0.2c$ is much larger than that of ${\it\delta}>0.2c$ . The lift coefficient increases by nearly 18.7 % at the lower end of the minimum distance range. Such a behaviour has a particular relevance for the studies of the fruit fly model (Liang & Sun Reference Liang and Sun2013), in which the ${\it\delta}$ is about $0.25c$ and the relative lift increase is about 7.2 %.

Figure 13.

Variation in (a) cycle-averaged lift and (b) relative overall lift enhancement for the WB model with changing the minimum wing–body distances ( ${\it\delta}$ ).

Figure 14 presents perspective views of the vortex topology and the spanwise vorticity slices for ${\it\delta}=0.2c$ and this can be examined in conjunction with the corresponding plot for ${\it\delta}=0.07c$ (figures 6 b and 9 a). The important feature that needs to be pointed out here is the absence of the PBVs and the significantly weakened TXVs compared to the flow at ${\it\delta}=0.07c$ . This clearly indicates that the minimum wing–body distance of the model will significantly affect the formation of PBVs and TXVs and the associated aerodynamic force.

Figure 14.

(a) 3D wake structures and (b) spanwise vorticity contours of the WB model at ${\it\delta}=0.2c$ .

Figure 15(a,b) show the variation in the cycle-averaged lift coefficient with body inclination angle and the corresponding relative lift increase at $k=0.83$ and ${\it\delta}=0.07c$ . It is worth noting that the increases of the body inclination angle lead to the monotonically increase of the lift production on wings but has different influence on bodies, especially for the WB models. The maximum relative lift increase was found at ${\it\chi}=20^{\circ }$ . The body inclination angle of the cicada forward flight in our experimental observation is close to this optimal value.

Figure 15.

Variation in (a) cycle-averaged lift and (b) relative overall lift enhancement for the WB and WN/BD models with changing body inclination angle ( ${\it\chi}$ ).

For the reduced frequency study, it is useful to examine the effect of this parameter on the lift production. Figure 16 shows the variation of mean lift versus reduced frequency of the flapping wings at ${\it\delta}=0.07c$ and ${\it\chi}=28^{\circ }$ . The plot shows that at the lower end of the reduced frequency the lift enhancement of the body is much stronger than that at the high end of the reduced frequency. It shows that the maximum relative lift increase (24.7 %) occurs at $k=0.75k_{0}$ and it decreases after $k>0.75k_{0}$ . This implies that larger size cicadas with slightly smaller flapping frequency are prone to utilize this lift enhancement mechanism for additional lift gain due to WBIs.

Figure 16.

Variation in (a) cycle-averaged lift and (b) relative overall lift enhancement for the WB and WN/BD models with changing the reduced frequency ( $k$ ). In which, the unit of the horizontal axis ( $\times k_{0}$ ) represents the original reduced frequency, where $k_{0}=0.83$ .