A computational study of the aerodynamics and forewing-hindwing interaction of a model dragonfly in forward flight

Scientists have always been fascinated by the flight of dragonflies. Analysis based on quasi-steady aerodynamic theory has shown that the vertical force required for weight support is much greater than the steady-state values measured from dragonfly wings, suggesting that unsteady aerodynamics must play important roles in the flight of dragonflies(Norberg, 1975; Wakeling and Ellington, 1997a,b,c).

Force measurement on a tethered dragonfly was conducted by Somps and Luttges (1985). It was shown that over some part of a stroke cycle, vertical force was many times larger than the dragonfly weight. They considered that the large force might be due to the effect of forewing-hindwing interaction. Flow visualization studies on flapping model dragonfly wings were conducted by Saharon and Luttges(1988, 1989), and it was shown that constructive or destructive wing/flow interactions might occur, depending on the kinematic parameters of the flapping motion. In these studies, only the total force of the fore- and hindwings was measured and, moreover, force measurements and flow visualizations were conducted in separate works. Experimental (Freymuth, 1990)and computational (Wang, 2000)studies on an airfoil (two-dimensional wing) in dragonfly hovering mode showed that large vertical force was produced during each downstroke and that the mean vertical force was enough to support the weight of a typical dragonfly. During each downstroke, a vortex pair was created; the large vertical force was explained by the downward two-dimensional jet induced by the vortex pair(Wang, 2000). In these works(Freymuth, 1990; Wang, 2000), because only a single airfoil was used, the effects of interaction between the fore- and hindwings and the three-dimensional flow effects could not be considered. Flow visualization studies on free-flying and tethered dragonflies were recently conducted by Thomas et al.(2004). It was shown that dragonflies fly by using unsteady aerodynamic mechanisms to generate leading-edge vortices (LEVs) or high lift when needed and that the dragonflies controlled the flow mainly by changing the angle of attack of the wings. Their results represent the only existing data on the flow around the wings of free-flying dragonflies.

Recently, Sun and Lan(2004) studied the aerodynamics and the forewing-hindwing interaction of the dragonfly Aeshna juncea in hover flight, using the method of computational fluid dynamics(CFD). Three-dimensional wings and wing kinematics data of free-flight were employed in the study. They showed that the vertical force coefficient of the forewing or the hindwing was twice as large as the quasi-steady value and that the mean vertical force could balance the dragonfly weight. They also showed that the large vertical force coefficient was due to the LEV associated with the delayed stall mechanism and that the interaction between the fore- and hindwings was not very strong and was detrimental to the vertical force generation. The result of detrimental interaction is interesting. But Sun and Lan (2004) investigated only a specific case of flight in Aeshna juncea, i.e. hovering with 180°phase difference between the fore- and hindwings. Whether the result that forewing-hindwing interaction is detrimental is a local result due to the specific kinematics used or is a more general result is not known. It is desirable to make further studies on dragonfly aerodynamics at various flight conditions and on the problem of forewing-hindwing interaction.

In the present study, we address the above questions by numerical simulation of the flows of a model dragonfly in forward flight. The vertical force and thrust are made to balance the insect weight and body-drag,respectively, by adjusting the angles of attack of the wings, so that the simulated flight could better approximate the real flight. The phasing and the incoming flow speed (flight speed) of the model dragonfly are systematically varied. At each flight speed, four phase differences -0°, 60°, 90°and 180° (the hindwing leads the forewing motion) - are considered. Dragonflies vary the phase difference between the fore- and hindwings with different behaviours (Norberg,1975; Azuma and Watanabe,1988; Reavis and Luttges,1988; Wakeling and Ellington,1997b; Wang et al.,2003; Thomas et al.,2004). It has been shown that a 55-100° phase difference (the hindwing leads forewing motion) is commonly used in straight forward flight(e.g. Azuma and Watanabe, 1988;Wang et al., 2004) and a 180° phase difference is used in hovering (e.g. Norberg, 1975). Recent observation by Thomas et al.(2004) has shown that 180°phase difference is also used in forward flight. We chose 60°, 90° and 180° to represent the above range of phase difference. Although 0°phase difference (parallel stroking) has been mainly found in accelerating or manoeuvring flight (e.g. Alexander,1986; Thomas et al.,2004), this phase difference is also included for reference. As in Sun and Lan (2004), the approach of solving the flow equations over moving overset grids is employed because of the unique feature of the motion, i.e. the fore- and hindwings move relative to each other.

The model fore- and hindwings (Fig. 1) are the same as those used in Sun and Lan(2004). The thickness of the wings is 1% of c (where c is the mean chord length of the forewing). The planforms of the wings are similar to those of the wings of Aeshna juncea (Norberg,1972). The fore- and hindwings are the same length, but the chord length of the hindwing is larger than that of the forewing. The radius of the second moment of the forewing area is denoted by r₂, and r₂=0.61R, where R is the wing length(the mean flapping velocity at r₂ is used as the reference velocity in the present study).

The flow equations and computational method used in the present study are the same as those used in Sun and Lan(2004). Only an outline of the method is given here. The Navier-Stokes equations are numerically solved using moving overset grids. The algorithm was first developed by Rogers and Kwak(1990) and Rogers et al.(1991) for single-grid, which is based on the method of artificial compressibility, and it was extended by Rogers and Pulliam (1994) to overset grids. The time derivatives of the momentum equations are differenced using a second-order, three-point backward difference formula. The derivatives of the viscous fluxes in the momentum equation are approximated using second-order central differences. For the derivatives of convective fluxes,upwind differencing based on the flux-difference splitting technique is used. A third-order upwind differencing is used at the interior points, and a second-order upwind differencing is used at points next to boundaries. With overset grids (Fig. 2), for each wing there is a body-fitted curvilinear grid, which extends a relatively short distance from the body surface, and in addition, there is a background Cartesian grid, which extends to the far-field boundary of the domain. The solution method for single-grid is applied to each of these grids; data are interpolated from one grid to another at the inter-grid boundary points.

Only the flow on the right of the plane of symmetry(Fig. 1A) is computed; the effects of left wings are taken into consideration by the central mirroring condition. The overset-grid system used here is the same as that in Sun and Lan (2004). Each of the wing grids had dimensions 29×77×45 in the normal direction, around the wing and in the spanwise direction, respectively, and the background grid had dimensions 46×94×72 in the Y-direction and directions parallel and normal to the stroke-planes, respectively. The time step value used (Δτ=0.02) is also the same as that in Sun and Lan(2004).

Conventionally, reference velocity used in the definition of force coefficients of a wing is the relative velocity of the wing. In the above definition of force coefficients, U is used as the reference velocity. At hovering, U is the mean relative velocity of the wings. It should be noted that at forward flight, U is not the mean relative velocity of the wings and the above definition of force coefficients is different from the conventional one.

In the flapping motion described above, we need to specify the flapping period (τ_c), the reference velocity (U), the geometrical angles of attack (α_d and α_u),the wing rotation duration (Δτ_r), the phase difference(γ_d) between hindwing and forewing, the mean flapping angle(⁠

For the dragonfly Aeshna juncea in hovering flight, the following kinematic data are available (Norberg,1975): β≈60°, n=36 Hz and Φ=69° for both wings;

Now, the only kinematic parameters left to be specified areα _d and α_u. In the present study,α _d and α_u are not treated as known input parameters but are determined in the calculation process; they are chosen such that the computed mean vertical force of the wings approximately equals the insect weight and the computed mean thrust approximately equals the body drag. The mean vertical force coefficient required for balancing the weight(C_V,W) is defined as C_V,W=mg/0.5ρU²(S_f+S_h);the body-drag coefficient (C_D,b) is defined as C_D,b=body-drag/0.5ρU²(S_f+S_h). Using the above data, C_V,W is computed as C_V,W=1.35. The body-drag of Aeshna juncea is not available. Here, the body-drag coefficients for dragonfly Sympetrum sanguineum (Wakeling and Ellington,1997a) are used (converted to the current definition of C_D,b). Values of C_D,b at various J are shown in Table 1.

In the present study, six advance ratios (J=0, 0.15, 0.30, 0.45,0.60, 0.75; V_∞=0-3.1 m s^-1) and, at each J, four phase differences (γ_d=180°, 90°,60° and 0°; hindwing leads the forewing motion), are considered. At a given set of J and γ_d, α_d andα _u are chosen such that the C_Vapproximately equals C_V,W, and C_Tapproximately equals C_D,b. The calculation procedure is as follows. At a given J and γ_d, a set of values ofα _d and α_u is estimated (how the starting values are estimated is described below). The flow equations are solved and the corresponding C_V and C_T are calculated. C_V is compared with C_V,W(1.35) and C_T is compared with C_D,b(Table 1). If C_V is different from C_V,W, or C_T is different from C_D,b,α _d and α_u are adjusted. The calculations are repeated until the difference between C_V and C_V,W is less than 0.05 and the difference between C_T and C_D,b is less than ∼0.01 (as will be seen below, in most cases, a difference between C_Tand C_D,b of less than 0.005 is achieved).

The case of J=0 (γ_d=180°) is computed first. For this case, values of α_d and α_u close to the real ones are available from Norberg(1975). For dragonfly Aeshna juncea hovering with γ_d=180°, Norberg(1975) observed that in the mid-portion of the downstroke, the wing chord was almost horizontal, and in the mid-portion of the upstroke it was close to the vertical; that is the real values of α_d and α_u should be around 50°and 20°, respectively (note that β=52°).α _d=50° and α_u=15° are used as the starting values, and the converged values of α_d andα _u are 52° and 8°, respectively. Using starting values that are not far from the real values can reduce the number of iterations. More importantly, there could be more than one solution due to the nonlinearity in aerodynamic force production, and by so doing, the calculation can generally converge to the realistic solution. Second, the case of J=0.15 (γ_d=180°) is computed, using the converged values of α_d and α_u of J=0 (γ_d=180°) as the starting values. Since J is not changed greatly, it is expected that these starting values are not very different from the realistic solution. The same is done,sequentially, for the cases of J=0.3, 0.45, 0.6 and 0.75(γ_d=180°). Next, the case of J=0(γ_d=90°) is computed, using the converged values ofα _d and α_u at J=0(γ_d=180°) as the starting values; then the cases of J=0.15-0.75 (γ_d=90°) are computed in the same way as in the corresponding cases of γ_d=180°. Finally,the cases of J=0-0.75, γ_d=60° and 0° are treated in a similar way.

The calculated results of α_d and α_u are shown in Table 2. Since, in each of the cases, the starting values of α_d andα _u are expected to be not far from the real values, it is reasonable to expect that these solutions are realistic. Let's examine how the calculated α_d and α_u vary with advance ratio, which can give some information on whether or not the solutions are realistic. As seen in Table 2,at a given γ_d, when J is increased,α _d decreases and α_u increases. This should be the correct trend of variation for the following reasons. When Jis increased, in the downstroke the relative velocity of the wing increases and, to keep the total vertical force from increasing (vertical force is mainly produced during the downstroke and it needs to be equal to the weight of the dragonfly), α_d should decrease; in the upstroke, the relative velocity decreases and, to produce enough thrust (thrust is mainly produced during the upstroke and a larger thrust is needed as J is increased), α_u should increase. As also seen in Table 2, α_uincreases with J at a relatively higher rate (α_uincreases approximately from 8° to 65° when J changes from 0 to 0.75). This is reasonable because, if α_u does not increase with J fast enough, the effective angle of attack of the wing would become negative (generally, operating at negative effective angle of attack is not realistic). The variations of α_d and α_uwith J also show that it is reasonable to expect that the solutions are realistic.

In Table 2, the mean total force coefficients (C̄_V, C̄_T), and the mean force coefficients of the forewing(C̄_V,f, C̄_T,f) and hindwing(C̄_V,h, C̄_T,h) are also given(C̄_V,f, C̄_T,f, etc. could show how much aerodynamic force is produced by the forewing or by the hindwing). C̄_V is close to C_V,W and C̄_T is closed to C_D,b, as they should be. The mean thrust (the body-drag)is much smaller than the mean vertical force (the weight); e.g. at J=0, 0.3 and 0.6, C̄_T is only 0, 1.4 and 6.6%of C̄_V, respectively. At a given J, α_d and α_u do not change greatly when γ_d is varied. For example, at J=0.15,α _d and α_u are 44° and 14°,respectively, at γ_d=180°; 42° and 13.2° atγ _d=90°; 40° and 12.5° atγ _d=60°; 38° and 9.7° atγ _d=0°.

The fact that changing γ_d from 180° to 0° does not influence α_d and α_u values greatly indicates that the forewing-hindwing interaction might not be very strong. This is because the interaction between the wings is expected to be sensitive to the relative motion, or to the phase difference, between the wings, and if strong interaction exits, the values of α_d and α_uwould be greatly influenced by varying γ_d from 180° to 0°.

Fig. 3 gives the time courses of C_V and C_T in one cycle for various forewing-hindwing phase differences for hovering flight(J=0). For a clear description of the time courses of the forces and flows, we express time during a cycle as a non-dimensional parameter, t̂, such that t̂=0 at the start of the downstroke of the hindwing and t̂=1 at the end of the following upstroke. At γ_d=180°, there are two large C_V peaks in one cycle, one in the first half-cycle(t̂=0-0.5) and the other in the second half-cycle (t̂=0.5-1.0) [this case has been investigated in Sun and Lan(2004) and is included here for comparison]. When the phase difference is changed toγ _d=90°, these two peaks merge into a large C_V peak between t̂=0 and t̂=0.75. The result atγ _d=60° is similar to that at γ_d=90°,except that the C_V peak is between t̂=0 and t̂=0.62 and is higher. For the case ofγ _d=0°, the C_V peak is between t̂=0 and t̂=0.5 and is even higher. C_V is the sum of C_V,f and C_V,h. Fig. 4 gives the time courses of C_V,f and C_V,h for the above cases. In all these cases, the hindwing produces a large C_V,h peak during its downstroke and a very small C_V,h during its upstroke; this is also true for the forewing. At γ_d=180°, the downstroke of the hindwing is in the first half-cycle (t̂=0-0.5)and the downstroke of the forewing is in the second half-cycle(t̂=0.5-1.0), resulting in the two C_V peaks (one between t̂=0 and t̂=0.5 and the other between t̂=0.5 and t̂=1.0; see the C_Vcurve for γ_d=180° in Fig. 3). Atγ _d=90°, the downstroke of the hindwing is still in the first half-cycle (between t̂=0 and t̂=0.5), but the downstroke of the forewing is between t̂=0.25 and t̂=0.75, resulting in the C_V peak between t̂=0 and t̂=0.75 (see the C_Vcurve for γ_d=90° in Fig. 3). The C_V peak for the cases of γ_d=60° and 0° in Fig. 3 can be explained similarly.

Fig. 5 gives the C_V and C_T results for forward flight at J=0.3. The effects of varying the phasing are similar to those in the cases of J=0, i.e. when γ_d is decreased from 180° to 90° (and below), the two C_V peaks (between t̂=0 and t̂=0.5 and between t̂=0.5 and t̂=1.0, respectively) merge into one C_V peak. This is generally true for other advance ratios considered.

Fig. 6 gives the time courses of C_V and C_T in one cycle for various advance ratios. For clarity, only the C_V and C_T curves for J=0, 0.3 and 0.6 are plotted [the C_V (or C_T) curve for J=0.15 is between those of J=0 and 0.3; the C_V (or C_T) curve for J=0.45 is between those of J=0.3 and 0.6; and the C_V (or C_T) curve for J=0.75 is close to that for J=0.6].

At γ_d=180° (Fig. 6A), as J is increased, the distributions of C_V in the first half-cycle(t̂=0-0.5) change greatly: C_V between t̂=0 and t̂=0.3 is decreased and C_V around t̂=0.4 is increased. As discussed above, C_V in the first half-cycle is due to the hindwing downstroke. The decrease in C_Vbetween t̂=0 and t̂=0.3 is caused mainly by two factors;(1) α_d of the hindwing is smaller at higher speeds(Table 2) and (2) at higher speeds, the forewing-hindwing interaction decreases the vertical force on the hindwing in this period (see below). The large increase in C_V around t̂=0.4 is due to the effect of pitching-up rotation of the hindwing. It is known that when a wing pitches up in an incoming flow, large aerodynamic forces could be produced; the higher the incoming flow speed, the larger the forces(Dickinson et al., 1999; Lan and Sun, 2001; Sun and Tang,2002). The hindwing undergoes pitching-up rotation at t̂=0.4. At higher J, the relative velocity is larger and, in addition, the portion of wing area behind the rotation-axis is relatively large for the hindwing (see Fig. 1A), resulting in the large C_V around t̂=0.4.

At γ_d=90°, 60° and 0°(Fig. 6C, E and G,respectively), the effects of increasing J on C_Vare similar to those in the case of γ_d=180°.

The vertical force coefficient of a wing is related to the lift and drag coefficients (see Eqn 1). Fig. 7 shows the vertical force, lift and drag coefficients of the hindwing and the forewing,respectively, for the case of J=0.3 and γ_d=180°. Fig. 8 shows the corresponding results for the case of J=0.6 and γ_d=180°. It is seen that for the forewing or the hindwing, the drag coefficient is larger than, or close to, the lift coefficient. Furthermore, β is large(52°). As a result (see Eqn 1), a large part of the vertical force coefficient is contributed by the drag coefficient. This is also true for other flight conditions. Our computations show that for all cases considered in the present study, 55-67%of the total vertical force is contributed by the drag of the wings. The results here are for hovering and forward flight conditions. For hovering,similar results have been obtained previously: Sun and Lan(2004) showed that for the same dragonfly as in the present study, 65% of the weight-supporting force is contributed by the wing drag; Wang(2004), using two-dimensional model, showed that a dragonfly might use drag to support about three-quarters of its weight.

Here, we present flows around the forewing and the hindwing for six representative cases: γ_d=180° and J=0, 0.3 and 0.6; γ_d=60° and J=0, 0.3 and 0.6. Figs 9, 10, 11 show the contours of the non-dimensional spanwise component of vorticity at half-wing length at various times of the stroke cycle, for the cases J=0, 0.3 and 0.6 ofγ _d=180°; Figs 12, 13, 14 show the corresponding results for the cases of γ_d=60°. Since the variation in J causes considerable changes in α_d andα _u, to guard against possible misinterpretation of the results, in each of Figs 9, 10, 11, 12, 13, 14, α_d andα _u are specified at the same time as J (this is also done in Fig. 15). In Figs 9, 10, 11, 12, 13, 14, τ₁,τ ₂ and τ₃ represent the times at 0.1τ_c after the start of the downstroke, the mid-downstroke and 0.4τ_c after the start of the downstroke of a wing,respectively; τ₄, τ₅ and τ₆represent the corresponding times of the upstroke of the wing.

First, we examine the cases of γ_d=180°. At J=0 for the forewing (Fig. 9A), during the downstroke a LEV of large size appears (see plots at τ₂ and τ₃ in Fig. 9A); during the upstroke,there is no LEV and the vorticity layers on the upper and lower surfaces of the wing are approximately the same (see plots at τ₅ andτ ₆ in Fig. 9A),indicating that the effective angle of attack is close to zero. For the hindwing (Fig. 9B), during the downstroke the flows are generally similar to those of the forewing, except that the LEV is a little smaller and a vortex layer shed from the trailing edge (trailing-edge vortex layer) of the forewing is around the hindwing at its mid-upstroke (see plot at τ₅ in Fig. 9B). At J=0.3(Fig. 10), the LEVs of the wings during their downstrokes are smaller than those at J=0 (compare Fig. 10 with Fig. 9); in fact, the LEV of the hindwing has the form of a thick vortex layer (see plots atτ ₂ and τ₃ in Fig. 10B), indicating that the flow is effectively attached. Another difference is that the trailing-edge vortex layer of the forewing is less close to the hindwing at its mid-upstroke than in the case of J=0 (comparing the plot at τ₅ in Fig. 10B with the plot atτ ₅ in Fig. 9B). At J=0.6 (Fig. 11),the LEVs of both the forewing and hindwing during their downstrokes have the form of a thick vortex layer (see plots at τ₂ andτ ₃ in Fig. 11Aand Fig. 11B), indicating that flows are effectively attached. The flow attachment during the downstrokes at relatively large J can be clearly seen from the sectional streamline plots shown in Fig. 15: as J increases, flows around the forewing and hindwing become more and more attached.

Next, we examine the cases of γ_d=60° (Figs 12, 13, 14). The flows vary with J in the same way as in the cases of γ_d=180°discussed above; that is, as J increases, the LEVs on the forewing and the hindwing downstrokes decease in size (becoming a vortex layer at relatively large J), and the hindwing in its downstroke meets less and less of the trailing-edge vortex layer of the forewing (compare Figs 12, 13 and 14). At a given J,the flows of the fore- and hindwings are not greatly different from those in the case of γ_d=180°, except that the hindwing in its upstroke meets the trailing-edge vortex layer of the forewing at an earlier time (compare Figs 12, 13 and 14 with Figs 9, 10 and 11, respectively). The fact that there do not exist large differences between the flows forγ _d=60° and γ_d=180° indicates that the forewing-hindwing interaction might not be very strong.

Figs 16, 17, 18, 19 compare the time courses of C_V,sf, C_V,sh, C_T,sf and C_T,sh with those of C_V,f, C_V,h, C_T,fand C_T,h, respectively. The differences between C_V,sf and C_V,f, etc., show the interaction effects. At a given γ_d and J (e.g.γ _d=180° and J=0.6; Fig. 16E), the vertical force coefficient of a wing is decreased at certain periods and increased at some other periods of a cycle due to forewing-hindwing interaction. When Jis varied (e.g. comparing Fig. 16A,C,E) or γ_d is varied (e.g. comparing Figs 16A, 17A and 18A), the interaction effect occurs at different periods of the cycle and its strength may change. This is because, at a given time in the stroke cycle, a wing is at a different position relative to the wake of the other wing when J orγ _d is varied.

Recently, Maybury and Lehmann(2004) conducted experiments on interaction between two robotic wings. In their experiment, the two wings are stacked vertically (forewing on the top), the stroke planes are horizontal and the wings operate in still air. Although their experimental set-up is different from the set-up of our simulation, there is some resemblance between their experiment and our hovering simulation: the hindwing operates in the wake of the forewing and the forewing is also influenced by the disturbed flow due to the hindwing. Thus, the results on interaction effects obtained by these two studies might be similar to some extent. Data in fig. 3D of Maybury and Lehmann (2004) show that between a phase shift of 0 and 50% of the stroke cycle(γ_d≈0-180°), the total vertical force is reduced by approximately 6-16% due to the interaction. The results in the present study show that between γ_d≈60-180°, the total vertical force is reduced by 7.8-15% due to the interaction (see Table 3, J=0).

Results in the present computations (24 cases of different phasing and advance ratios) show that for the forewing or the hindwing, although its vertical force coefficient at certain periods of the stroke cycle can be slightly increased by the forewing-hindwing interaction effects, its mean vertical force coefficient is decreased by the interaction effects. That is,the forewing-hindwing interaction is detrimental to the vertical force generation (and also to the resultant force generation; as mentioned above,vertical force is very close to the resultant force because the thrust is much smaller than the vertical force). This is remarkable but not totally unexpected. For all the cases considered, each of the fore- and hindwings produces a mean vertical force coefficient close to half that needed to support the insect weight (see C̄_V,f and C̄_V,h in Table 2). In producing an upward force, a downward flow must be generated. Thus, in general, a wing would move in the downwash-velocity field induced by the other wing, reducing its vertical force.

Somps and Luttges (1985),based on their experiments, suggested that forewing-hindwing interaction might enhance aerodynamic force production. Results in the present study, however,show that the interaction is detrimental. It is of interest to discuss the present results in relation to those of Somps and Luttges(1985). In their experiment with a tethered dragonfly (in still air; wings flapping withγ _d≈80°), Somps and Luttges(1985) measured the time course of the total vertical force, which has a single large peak in each cycle (see fig. 2c of Somps and Luttges,1985); the mean vertical force is more than twice the body weight. Based on the fact that one single large vertical force peak is produced in each cycle (rather than the double peaks they expected from the sum of the forces produced independently by the fore- and hindwings), they considered that the forewing-hindwing interaction must be strong and suggested that it played an important role in generating the large vertical force. Our vertical force time histories for γ_d=60° and 90° at hovering are very similar to those in Somps and Luttges(1985), also having a single large peak in each cycle [compare the C_V curve forγ _d=60° or 90° in Fig. 3A with the curve in fig. 2c of Somps and Luttges(1985)]. However, analyses in the present study clearly show that the large single force peak is not due to forewing-hindwing interaction but rather to the overlap of the single force peak produced by the hindwing with that by the forewing.

As seen in Figs 9, 10, 11, 12, 13, 14, 15, at hovering(J=0), flows on both the forewing and hindwing during the loaded downstroke are separated and large LEVs exist. As J increases, the LEVs become smaller and smaller and the flows become more and more attached. The flows of the hindwing downstroke are effectively attached at J=0.3 and those of the forewing downstroke are effectively attached at J=0.6 (see e.g. Fig. 15). That is, in producing the aerodynamic forces needed for flight, the model dragonfly uses separated flows with LEVs at hovering and low J, uses both separated and attached flows at medium J, and uses attached flow at high J.

At hovering and low J, the relative velocity of a wing is mainly due to the flapping motion and is relatively low. Thus, high `aerodynamic force coefficients' are needed (in the present section, aerodynamic force coefficients are coefficients defined in the conventional way; that is, the reference velocity used is the relative velocity of the wing; note that reference velocity used in the definition of the aerodynamic force coefficients in the proceeding sections is U, which is smaller than the relative velocity of the forewing or the hindwing in the case of forward flight). The dragonfly must use the separated flows with LEVs to generate the high aerodynamic force coefficients.

At high J, the relative velocity is contributed by both the flapping motion and the relatively high forward velocity and is relatively high. Thus, relatively low aerodynamic force coefficients are needed. The dragonfly does not need to use separated flows; instead, it uses attached flows. As an example, we estimate the mean relative velocity of a section of the forewing (or hindwing) at a distance r₂ from the wing root at J=0.6. Using the diagram in Fig. 20, the relative velocity is estimated as 1.78U [U is the mean relative velocity of this section at hovering (J=0)]. The mean relative velocity is 1.78 times as large as that at hovering, and the vertical force coefficient needed would be about one-third of that needed for hovering. Therefore, at J=0.6, attached flows could produce the required aerodynamic force coefficients.

Recently, Thomas et al.(2004) presented flow visualization results for free-flying and tethered dragonflies. Some of their visualization tests were made for the dragonfly Aeshna mixta flying freely at V_∞=1.0 m s^-1 (see, for example,fig. 6 of Thomas et al.,2004). Their results show that the dragonfly uses counter-stroking(γ_d=180°), with an LEV on the forewing downstroke and attached flow on the hindwing down- and upstrokes. The model dragonfly in the present study is modelled using the available morphological and kinematic data of the dragonfly Aeshna juncea, which is of the same genus as the dragonfly in the experiment. Moreover, in the flight of the model dragonfly,force-balance conditions are satisfied, and the flight could be a good approximation of the real flight. Therefore, we can make comparisons between the computed and experimental results. At U=0.3, V_∞ of the model dragonfly is 1.23 m s^-1,close to that in the experiment. Our results show that at this flight velocity there is a LEV on the forewing downstroke and the flows on the hindwing down-and upstrokes are approximately attached (Figs 10B, 15), in agreement with the flow visualization results of the free-flying dragonfly.

The above comparison is for an intermediate advance ratio. For high and very low advance ratios, there are also similarities between the visualizations of Thomas et al.(2004) and the simulation of the present study. Based on two available free flight sequences, Thomas et al.(2004) suggested (p. 4308)that at fast flight (high advance ratio), flows on the forewing and the hindwing were both attached; our results show that at J=0.6 (Figs 11, 14), the flows on both the forewing and the hindwing are approximately attached. At very low speed, they showed (video S2 in their supplementary material) that flows were separated on the hindwing as well as on the forewing; our simulation gives similar results(Figs 9, 12).

List of symbols

 
• c

  mean chord length of forewing

 
• C_D,b

  body-drag coefficient

 
• C_d,f

  drag coefficient of forewing

 
• C_d,h

  drag coefficient of hindwing

 
• C_l,f

  lift coefficient of forewing

 
• C_l,h

  lift coefficient of hindwing

 
• C̄ _T

  mean total thrust coefficient

 
• C_T

  total thrust coefficient

 
• C_T,f

  thrust coefficient of forewing

 
• C_T,h

  thrust coefficient of hindwing

 
• C_T,sf

  thrust coefficient of single forewing

 
• C_T,sh

  thrust coefficient of single hindwing

 
• C̄ _V

  mean total vertical force coefficient

 
• C_V

  total vertical force coefficient

 
• C_V,f

  vertical force coefficient of forewing

 
• C_V,h

  vertical force coefficient of hindwing

 
• C_V,NI

  total vertical force coefficient without interaction

 
• C̄ _V,NI

  mean total vertical force coefficient without interaction

 
• C_V,sf

  vertical force coefficient of single forewing

 
• C_V,sh

  vertical force coefficient of single hindwing

 
• ΔC_V

  percentage of increment in mean total vertical force coefficient due to forewing-hindwing interaction

 
• C_V,W

  mean vertical force required for balancing the weight

 
• d_f

  drag, forewing

 
• d_h

  drag, hindwing

 
• J

  advance ratio

 
• l_f

  lift, forewing

 
• l_h

  lift, hindwing

 
• m

  mass of the insect

 
• n

  flapping frequency

 
• O

  origin of the inertial frame of reference

 
• r

  radial position along wing length

 
• R

  wing length

 
• r₂

  radius of the second moment of wing area of forewing

 
• Re

  Reynolds number

 
• S_f

  area of one wing (forewing)

 
• S_h

  area of one wing (hindwing)

 
• t̂

  time

 
• t̂

  non-dimensional parameter expressing time during a cycle(t̂=0 at the start of the downstroke of the hindwing and t̂=1 at the end of the following upstroke)

 
• T

  total thrust

 
• T_f

  thrust of forewing

 
• T_h

  thrust of hindwing

 
• T_sf

  thrust of single forewing

 
• T_sf

  thrust of single hindwing

 
• U

  reference velocity

 
• u_t

  translational velocity of a wing

 
• u_t⁺

  non-dimensional translational velocity of a wing

 
• V̄

  mean total vertical force

 
• V

  total vertical force

 
• V_∞

  free-stream velocity or flight velocity

 
• V_f

  vertical force of forewing

 
• V_h

  vertical force of hindwing

 
• V_NI

  vertical force without interaction

 
• V_sf

  vertical force of single forewing

 
• V_sh

  vertical force of single hindwing

 
• X,Y,Z

  coordinates in inertial frame of reference (Z in vertical direction)

 
• \({\dot{{\alpha}}}\)

  angular velocity of flip rotation

 
• \({\dot{{\alpha}}}^{+}\)

  non-dimensional angular velocity of flip rotation

 
• \({\dot{{\alpha}}}_{0}^{+}\)

  a constant

 
• αd

  geometrical angle of attack of downstroke

 
• αu

  geometrical angle of attack of upstroke

 
• β

  stroke plane angle

 
• γ

  phase angle of the translation of a wing

 
• γd

  difference in phase angle between the hindwing and the forewing

 
• \({\dot{{\phi}}}\)

  angular velocity of azimuthal rotation

 
• π

  azimuthal or positional angle

 
• \({\bar{{\phi}}}\)

  mean flapping angle

 
• Φ

  stroke amplitude

 
• ν

  kinematic viscosity of the air

 
• ρ

  density of fluid

 
• τ

  non-dimensional time

 
• τc

  period of one flapping cycle (non-dimensional)

 
• τr

  time when pitching rotation starts (non-dimensional)

 
• Δτr

  duration of wing rotation or flip duration (non-dimensional)

We thank the two referees whose helpful comments and valuable suggestions greatly improved the quality of the paper. This research was supported by the National Natural Science Foundation of China (10232010, 10472008).