## SUMMARY

The aerodynamics and forewing-hindwing interaction of a model dragonfly in forward flight are studied, using the method of numerically solving the Navier-Stokes equations. Available morphological and stroke-kinematic parameters of dragonfly (*Aeshna juncea*) are used for the model dragonfly. Six advance ratios (*J*; ranging from 0 to 0.75) and, at each *J*, four forewing-hindwing phase angle differences(γ_{d}; 180°, 90°, 60° and 0°) are considered. The mean vertical force and thrust are made to balance the weight and body-drag, respectively, by adjusting the angles of attack of the wings, so that the flight could better approximate the real flight.

At hovering and low *J* (*J*=0, 0.15), the model dragonfly uses separated flows or leading-edge vortices (LEV) on both the fore- and hindwing downstrokes; at medium *J* (*J*=0.30, 0.45), it uses the LEV on the forewing downstroke and attached flow on the hindwing downstroke; at high *J* (*J*=0.6, 0.75), it uses attached flows on both fore- and hindwing downstrokes. (The upstrokes are very lightly loaded and, in general, the flows are attached.)

At a given *J*, at γ_{d}=180°, there are two vertical force peaks in a cycle, one in the first half of the cycle, produced mainly by the hindwing downstroke, and the other in the second half of the cycle, produced mainly by the forewing downstroke; atγ _{d}=90°, 60° and 0°, the two force peaks merge into one peak. The vertical force is close to the resultant aerodynamic force[because the thrust (or body-drag) is much smaller than vertical force (or the weight)]. 55-65% of the vertical force is contributed by the drag of the wings.

The forewing-hindwing interaction is detrimental to the vertical force (and resultant force) generation. At hovering, the interaction reduces the mean vertical force (and resultant force) by 8-15%, compared with that without interaction; as *J* increases, the reduction generally decreases (e.g. at *J*=0.6 and γ_{d}=90°, it becomes 1.6%). A possible reason for the detrimental interaction is as follows: each of the wings produces a mean vertical force coefficient close to half that needed for weight support, and a downward flow is generated in producing the vertical force; thus, in general, a wing moves in the downwash-velocity field induced by the other wing, reducing its aerodynamic forces.

## Introduction

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.

## Materials and methods

### The model wings

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 *r*_{2}, and *r*_{2}=0.61*R*, where *R* is the wing length(the mean flapping velocity at *r*_{2} is used as the reference velocity in the present study).

### The flow computation method and evaluation of the aerodynamic forces

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).

*l*

_{f}and

*d*

_{f}denote the lift and drag of the forewing, respectively;

*l*

_{h}and

*d*

_{h}denote the lift and drag of the hindwing, respectively. Resolving the lift and drag into the

*Z*and

*X*axes gives the vertical force and thrust of a wing.

*V*

_{f}and

*T*

_{f}denote the vertical force and thrust of the forewing, respectively;

*V*

_{h}and

*T*

_{h}denote the vertical force and thrust of the hindwing,respectively. For the forewing:

*V*

_{f},

*T*

_{f},

*V*

_{h},

*T*

_{h},

*l*

_{f},

*d*

_{f},

*l*

_{h}and

*d*

_{h}are denoted as

*C*

_{V,f},

*C*

_{T,f},

*C*

_{V,h},

*C*

_{T,h},

*C*

_{l,f},

*C*

_{d,f},

*C*

_{l,h}and

*C*

_{d,h}, respectively. They are defined as:

*S*

_{f}and

*S*

_{h}are the areas of the fore- and hindwings, respectively. The total vertical force (

*V*) and total thrust (

*T*) of the fore- and hindwings are

*V=V*

_{f}+

*V*

_{h}and

*T=T*

_{f}+

*T*

_{h}, respectively. The coefficients of

*V*and

*T*are denoted as

*C*

_{V}and

*C*

_{T}, respectively, and defined as:

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.

### Kinematics of flapping wings

*V*

_{∞}, and the stroke plane angle is denoted by β(Fig. 1B). The azimuthal rotation of a wing is called `translation', and the pitching (or flip)rotation of the wing near the end of a half-stroke and at the beginning of the following half-stroke is called rotation. The speed at

*r*

_{2}is called the translational speed. The wing translates downwards and upwards along the stroke plane and rotates during stroke reversal(Fig. 1B). The translational velocity is denoted by

*u*

_{t}and is given by:

*u*

_{t}

^{+}=

*u*

_{t}/

*U*(

*U*is the reference velocity); the non-dimensional timeτ=

*tU/c*(

*t̂*is the time;

*c*is the mean chord length of the forewing, used as reference length in the present study); τ

_{c}is the non-dimensional period of the flapping cycle; and γ is the phase angle of the translation of the wing. The reference velocity is

*U*=2Φ

*nr*

_{2}, whereΦ and

*n*are the stroke amplitude and stroke frequency of the forewing, respectively. Denoting the azimuth-rotational velocity as

_{d}for the downstroke and by α

_{u}for the upstroke (Fig. 1). Around the stroke reversal, the angle of attack changes with time, and the angular velocity (α) is given by:

_{r}is the time at which the rotation starts; andΔτ

_{r}is the time interval over which the rotation lasts. In the time interval of Δτ

_{r}, the wing rotates fromα

_{u}to α

_{d}. Therefore, whenα

_{d}, α

_{u}and Δτ

_{r}are specified,

_{u}to α

_{d}, and the sign of the right-hand side of Eqn 7 should be reversed). The axis of the flip rotation is located at a distance of 24% of the mean chord length of the wing from the leading edge. With

*U*and

*c*as the reference velocity and reference length, respectively, the Reynolds number (

*Re*) is defined as

*Re=Uc*/ν=2Φ

*nr*

_{2}

*c*/ν (ν is the kinematic viscosity of the air), and the advance ratio (

*J*) is defined as

*J=V*

_{∞}/2Φ

*nR*)=

*V*

_{∞}/(

*UR/r*

_{2}).

### Non-dimensional parameters of wing motion

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(

*Re*and

*J*.

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;

*m*) is 754 mg; forewing length is 4.74 cm; hindwing length is 4.60 cm; the mean chord lengths of the forewing and the hindwing are 0.81 cm and 1.12 cm, respectively. In the present study, we assume that for the dragonfly,

*n*and Φ do not vary with flight speed [data in Azuma and Watanabe(1988) show that

*n*hardly varies with flight speed and Φ is increased only at very high speed]. On the basis of the above data, we use the following parameters for the model dragonfly: the length of both wings (

*R*) is 4.7 cm(

*S*

_{f}and

*S*

_{h}are 3.81 and 5.26 cm

^{2}, respectively); the reference length (

*c*) is 0.81 cm;

*U*=2Φ

*nr*

_{2}=2.5 m s

^{-1};

*Re=Uc*/ν≈1350; τ

_{c}=

*U/nc*=8.58. Norberg(1975) did not provide the rate of wing rotation during stroke reversal. Reavis and Luttges(1988) made measurements on some dragonflies and it was found that maximum α was ∼10 000-30 000 deg. s

^{-1}. Here, α is set as 20 000 deg. s

^{-1},giving Δτ

_{r}=3.36. In hovering, the body of dragonfly

*Aeshna juncea*is horizontal(Norberg, 1975). We assume it is also horizontal at forward flight. The angle between the body axis and the stroke plane hardly changes (Azuma and Watanabe, 1988; Wakeling and Ellington, 1997b), therefore β at forward flight can be assumed to be the same as that at hovering [in Sun and Lan's(2004) study of hovering flight, β=52° was used; the same value is used here]. We also assume that at all speeds considered, geometrical angles of attack are the same for fore- and hindwings. In the present study, γ

_{d}and

*J*are varied systematically to study their effects, therefore they are known.

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}=*m g*/0.5ρ

*U*

^{2}(

*S*

_{f}+

*S*

_{h});the body-drag coefficient (

*C*

_{D,b}) is defined as

*C*

_{D,b}=body-drag/0.5ρ

*U*

^{2}(

*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.

J
. | 0 . | 0.15 . | 0.30 . | 0.45 . | 0.60 . | 0.75 . |
---|---|---|---|---|---|---|

V_{∞} (m s^{-1}) | 0.0 | 0.62 | 1.23 | 1.85 | 2.47 | 3.08 |

Re_{body} | 0.0 | 3115 | 6230 | 9346 | 12461 | 15576 |

C_{D,b} | 0.0 | 0.004 | 0.020 | 0.045 | 0.080 | 0.125 |

J
. | 0 . | 0.15 . | 0.30 . | 0.45 . | 0.60 . | 0.75 . |
---|---|---|---|---|---|---|

V_{∞} (m s^{-1}) | 0.0 | 0.62 | 1.23 | 1.85 | 2.47 | 3.08 |

Re_{body} | 0.0 | 3115 | 6230 | 9346 | 12461 | 15576 |

C_{D,b} | 0.0 | 0.004 | 0.020 | 0.045 | 0.080 | 0.125 |

*J,* advance ratio; *V*_{∞}, flight velocity; *Re*_{body}, body Reynolds number; *C*_{D,b},body-drag coefficient.

## Results

### Force balance in the flight

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*_{V}approximately equals *C*_{V,W}, and *C*_{T}approximately 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*_{T}and *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 *J*is 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_{u}increases with *J* at a relatively higher rate (α_{u}increases 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_{u}with *J* also show that it is reasonable to expect that the solutions are realistic.

γ_{d}
. | J
. | α_{d} (deg.)
. | α_{u} (deg.)
. | C̄ _{V,f}
. | C̄ _{V,h}
. | C̄ _{T,f}
. | C̄ _{T,h}
. | C̄ _{V}
. | C̄ _{T}
. |
---|---|---|---|---|---|---|---|---|---|

180° | 0 | 52.0 | 8.0 | 0.56 | 0.79 | 0.022 | -0.037 | 1.35 | -0.015 |

0.15 | 44.0 | 14.0 | 0.68 | 0.70 | 0.063 | -0.059 | 1.38 | 0.004 | |

0.30 | 36.0 | 22.0 | 0.66 | 0.69 | 0.066 | -0.049 | 1.35 | 0.017 | |

0.45 | 33.0 | 36.0 | 0.64 | 0.73 | 0.074 | -0.024 | 1.37 | 0.050 | |

0.60 | 32.0 | 51.0 | 0.60 | 0.72 | 0.077 | 0.008 | 1.32 | 0.085 | |

0.75 | 34.0 | 65.0 | 0.62 | 0.75 | 0.083 | 0.066 | 1.37 | 0.149 | |

90° | 0 | 52.0 | 7.0 | 0.54 | 0.78 | 0.023 | -0.020 | 1.32 | 0.003 |

0.15 | 42.0 | 13.2 | 0.62 | 0.74 | 0.076 | -0.074 | 1.36 | 0.002 | |

0.30 | 33.0 | 21.4 | 0.57 | 0.74 | 0.064 | -0.048 | 1.31 | 0.016 | |

0.45 | 31.0 | 35.3 | 0.57 | 0.77 | 0.049 | -0.004 | 1.34 | 0.045 | |

0.60 | 31.0 | 50.0 | 0.57 | 0.74 | 0.041 | 0.044 | 1.32 | 0.085 | |

0.75 | 33.0 | 64.0 | 0.58 | 0.75 | 0.055 | 0.096 | 1.33 | 0.152 | |

60° | 0 | 48.0 | 5.5 | 0.56 | 0.76 | 0.050 | -0.054 | 1.32 | -0.004 |

0.15 | 40.0 | 12.5 | 0.63 | 0.76 | 0.089 | -0.084 | 1.39 | 0.005 | |

0.30 | 32.0 | 21.8 | 0.57 | 0.75 | 0.076 | -0.058 | 1.32 | 0.018 | |

0.45 | 31.0 | 35.0 | 0.59 | 0.79 | 0.053 | -0.010 | 1.38 | 0.043 | |

0.60 | 31.0 | 50.0 | 0.59 | 0.74 | 0.049 | 0.040 | 1.33 | 0.089 | |

0.75 | 33.0 | 64.0 | 0.61 | 0.71 | 0.060 | 0.108 | 1.32 | 0.168 | |

0° | 0 | 45.0 | 6.8 | 0.60 | 0.75 | 0.138 | -0.134 | 1.35 | 0.004 |

0.15 | 38.0 | 9.7 | 0.65 | 0.71 | 0.124 | -0.120 | 1.36 | 0.004 | |

0.30 | 31.5 | 19.9 | 0.61 | 0.72 | 0.103 | -0.085 | 1.33 | 0.018 | |

0.45 | 31.5 | 35.0 | 0.63 | 0.71 | 0.091 | -0.028 | 1.34 | 0.063 | |

0.60 | 31.5 | 48.5 | 0.62 | 0.67 | 0.064 | 0.002 | 1.29 | 0.066 | |

0.75 | 35.0 | 63.0 | 0.72 | 0.66 | 0.072 | 0.086 | 1.38 | 0.158 |

γ_{d}
. | J
. | α_{d} (deg.)
. | α_{u} (deg.)
. | C̄ _{V,f}
. | C̄ _{V,h}
. | C̄ _{T,f}
. | C̄ _{T,h}
. | C̄ _{V}
. | C̄ _{T}
. |
---|---|---|---|---|---|---|---|---|---|

180° | 0 | 52.0 | 8.0 | 0.56 | 0.79 | 0.022 | -0.037 | 1.35 | -0.015 |

0.15 | 44.0 | 14.0 | 0.68 | 0.70 | 0.063 | -0.059 | 1.38 | 0.004 | |

0.30 | 36.0 | 22.0 | 0.66 | 0.69 | 0.066 | -0.049 | 1.35 | 0.017 | |

0.45 | 33.0 | 36.0 | 0.64 | 0.73 | 0.074 | -0.024 | 1.37 | 0.050 | |

0.60 | 32.0 | 51.0 | 0.60 | 0.72 | 0.077 | 0.008 | 1.32 | 0.085 | |

0.75 | 34.0 | 65.0 | 0.62 | 0.75 | 0.083 | 0.066 | 1.37 | 0.149 | |

90° | 0 | 52.0 | 7.0 | 0.54 | 0.78 | 0.023 | -0.020 | 1.32 | 0.003 |

0.15 | 42.0 | 13.2 | 0.62 | 0.74 | 0.076 | -0.074 | 1.36 | 0.002 | |

0.30 | 33.0 | 21.4 | 0.57 | 0.74 | 0.064 | -0.048 | 1.31 | 0.016 | |

0.45 | 31.0 | 35.3 | 0.57 | 0.77 | 0.049 | -0.004 | 1.34 | 0.045 | |

0.60 | 31.0 | 50.0 | 0.57 | 0.74 | 0.041 | 0.044 | 1.32 | 0.085 | |

0.75 | 33.0 | 64.0 | 0.58 | 0.75 | 0.055 | 0.096 | 1.33 | 0.152 | |

60° | 0 | 48.0 | 5.5 | 0.56 | 0.76 | 0.050 | -0.054 | 1.32 | -0.004 |

0.15 | 40.0 | 12.5 | 0.63 | 0.76 | 0.089 | -0.084 | 1.39 | 0.005 | |

0.30 | 32.0 | 21.8 | 0.57 | 0.75 | 0.076 | -0.058 | 1.32 | 0.018 | |

0.45 | 31.0 | 35.0 | 0.59 | 0.79 | 0.053 | -0.010 | 1.38 | 0.043 | |

0.60 | 31.0 | 50.0 | 0.59 | 0.74 | 0.049 | 0.040 | 1.33 | 0.089 | |

0.75 | 33.0 | 64.0 | 0.61 | 0.71 | 0.060 | 0.108 | 1.32 | 0.168 | |

0° | 0 | 45.0 | 6.8 | 0.60 | 0.75 | 0.138 | -0.134 | 1.35 | 0.004 |

0.15 | 38.0 | 9.7 | 0.65 | 0.71 | 0.124 | -0.120 | 1.36 | 0.004 | |

0.30 | 31.5 | 19.9 | 0.61 | 0.72 | 0.103 | -0.085 | 1.33 | 0.018 | |

0.45 | 31.5 | 35.0 | 0.63 | 0.71 | 0.091 | -0.028 | 1.34 | 0.063 | |

0.60 | 31.5 | 48.5 | 0.62 | 0.67 | 0.064 | 0.002 | 1.29 | 0.066 | |

0.75 | 35.0 | 63.0 | 0.72 | 0.66 | 0.072 | 0.086 | 1.38 | 0.158 |

γ_{d}, phase difference angle between fore- and hindwings; *J,* advance ratio; α_{d} and α_{u}, middle stroke angle of attack in the down- and upstrokes, respectively; *C̄*_{V} and *C̄*_{T}, mean total vertical and horizontal force coefficients, respectively; *C̄*_{V,f} and *C̄*_{T,f}, mean vertical and horizontal force coefficients of the forewing, respectively; *C̄*_{V,h} and *C̄*_{T,h}, mean vertical and horizontal force coefficients of the hindwing, respectively.

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_{u}would be greatly influenced by varying γ_{d} from 180° to 0°.

### The time courses of the aerodynamic forces

#### The effects of phasing

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*_{V}curve 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*_{V}curve 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.

#### The effects of flight speed

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*_{V}between *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*_{V}are similar to those in the case of γ_{d}=180°.

#### The lift and drag coefficients of the fore- and hindwings

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.

### The flows around the forewing and the hindwing

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, τ_{1},τ _{2} and τ_{3} 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; τ_{4}, τ_{5} and τ_{6}represent 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 τ_{2} and τ_{3} 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 τ_{5} 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 τ_{5} 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 τ_{3} 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 τ_{5} 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 τ_{2} 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.

### The forewing-hindwing interaction

*V*

_{sf}and

*T*

_{sf}, respectively; those for the single hindwing are denoted as

*V*

_{sh}and

*T*

_{sh}. The coefficients of

*V*

_{sf},

*T*

_{sf},

*V*

_{sh}and

*T*

_{sh}are denoted as

*C*

_{V,sf},

*C*

_{T,sf},

*C*

_{V,sh}and

*C*

_{T,sh}, respectively, and are defined as:

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,f}and *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 *J*is 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.

*V*

_{NI}) is the sum of

*V*

_{sf}and

*V*

_{sh}. The coefficient of

*V*

_{NI}is denoted as

*C*

_{V,NI}and defined as:

*C̄*

_{V,NI}be the mean value of

*C*

_{V,NI}. ThusΔ

*C*

_{V}=(

*C̄*

_{V}-

*C̄*

_{V,NI})/

*C̄*

_{V,NI}represents the percentage of increment in mean total vertical force coefficient due to the forewing-hindwing interaction (whenΔ

*C*

_{V}is negative, the interaction is detrimental to vertical force generation). The value of Δ

*C*

_{V}is given in Table 3. From the total vertical force and the total thrust, the total resultant force can be calculated. The increment in mean total resultant force coefficient due to the forewing-hindwing interaction is obtained in the same way as above, which is also given in Table 3. It is very close to Δ

*C*

_{V}. This is because, under the present flight conditions, the wings produce a much larger vertical force than thrust. As seen in Table 3, at all phase angles and advance ratios considered, the interaction is detrimental to the vertical force (or resultant force) generation. At hovering, the interaction reduces the mean total vertical force coefficient (or the mean total resultant force coefficient) by around 15% forγ

_{d}=180° and 90°, 8% for γ

_{d}=60°,and 3% for γ

_{d}=0°. As

*J*increases, forγ

_{d}=180°, 90° and 60°, the reduction decreases;but for γ

_{d}=0°, the reduction changes little from hovering to medium advance ratios (

*J*=0-0.3) and increases to 6-13% at higher advance ratios (

*J*=0.45-0.75).

. | Δ Ĉ_{V} (%). | . | . | . | |||
---|---|---|---|---|---|---|---|

J
. | γ_{d}=180°
. | γ_{d}=90°
. | γ_{d}=60°
. | γ_{d}=0°
. | |||

0 | -15 (-15.3) | -14.3 (-14.3) | -7.8 (-7.9) | -2.7 (-4.0) | |||

0.15 | -15 (-15.5) | -12.7 (-13.2) | -5.1 (-8.2) | -2.6 (-3.1) | |||

0.30 | -13.4 (-13.6) | -7.5 (-7.9) | -3.4 (-4.0) | -2.6 (-2.9) | |||

0.45 | -8.9 (-9.1) | -3.2 (-3.3) | 0 (-0.6) | -5.9 (-6.0) | |||

0.60 | -5.6 (-5.7) | -1.6 (-1.5) | 0 (-0.3) | -8.6 (-8.2) | |||

0.75 | -4.2 (-4.1) | -1.8 (-1.4) | -2.8 (-2.4) | -13.5 (-13.1) |

. | Δ Ĉ_{V} (%). | . | . | . | |||
---|---|---|---|---|---|---|---|

J
. | γ_{d}=180°
. | γ_{d}=90°
. | γ_{d}=60°
. | γ_{d}=0°
. | |||

0 | -15 (-15.3) | -14.3 (-14.3) | -7.8 (-7.9) | -2.7 (-4.0) | |||

0.15 | -15 (-15.5) | -12.7 (-13.2) | -5.1 (-8.2) | -2.6 (-3.1) | |||

0.30 | -13.4 (-13.6) | -7.5 (-7.9) | -3.4 (-4.0) | -2.6 (-2.9) | |||

0.45 | -8.9 (-9.1) | -3.2 (-3.3) | 0 (-0.6) | -5.9 (-6.0) | |||

0.60 | -5.6 (-5.7) | -1.6 (-1.5) | 0 (-0.3) | -8.6 (-8.2) | |||

0.75 | -4.2 (-4.1) | -1.8 (-1.4) | -2.8 (-2.4) | -13.5 (-13.1) |

*J,* advance ratio; γ_{d}, phase difference angle between fore- and hindwings; Δ*Ĉ*_{V}, percentage of increment in mean total vertical force coefficient due to interaction(negative number means interaction is detrimental). The numbers in parentheses are the corresponding results of the mean total resultant force coefficient.

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).

## Discussion

### The forewing-hindwing interaction is detrimental to the vertical force generation

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.

### Separated and attached flows

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 *r*_{2} from the wing root at *J*=0.6. Using the diagram in Fig. 20, the relative velocity is estimated as 1.78*U* [*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.

### Comparison with flow visualization results of free-flying dragonflies

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

- Δ
*C*_{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

- r
_{2}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)

## Acknowledgements

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).

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