Lift and power requirements of hovering flight in Drosophila virilis

The lift and power requirements of hovering flight in insects were systematically studied by Weis-Fogh(1972, 1973) and Ellington(1984c) using methods based on steady-state aerodynamic theory. It was shown that the steady-state mechanism was inadequate to predict accurately the lift and power requirements of small insects and of some large ones (Ellington,1984c).

In the past few years, much progress has been made in revealing the unsteady high-lift mechanisms of insect flight. Dickinson and Götz(1993) measured the aerodynamic forces on an aerofoil started impulsively at a high angle of attack [in the Reynolds number (Re) range of a fruit fly wing, Re=75-225]and showed that lift was enhanced by the presence of a dynamic stall vortex,or leading-edge vortex (LEV). Lift enhancement was limited to only 2-3 chord lengths of travel because of the shedding of the LEV. For most insects, a wing section at a distance of 0.5R (where R is wing length) from the wing base travels approximately 3.5 chord lengths during an up- or downstroke in hovering flight (Ellington,1984b). A section in the outer part of the same wing travels a larger distance, e.g. a section at 0.75R from the wing base travels approximately 5.25 chord lengths, which is much greater than 2-3 chord lengths(in forward flight, the section would travel an even larger distance during a downstroke).

Ellington et al. (1996)performed flow-visualization studies on a hawkmoth Manduca sextaduring tethered forward flight (forward speed in the range 0.4-5.7 m s^-1) and on a mechanical model of the hawkmoth wings that mimicked the wing movements of a hovering Manduca sexta [Re≈3500;in the present paper, Re for an insect wing is based on the mean velocity at r₂ (the radius of the second moment of wing area) and the mean chord length of the wing]. They discovered that the LEV on the wing did not shed during the translational motion of the wing in either the up- or downstroke and that there was a strong spanwise flow in the LEV.(They attributed the stabilization of the LEV to the effect of the spanwise flow.) The authors suggested that this was a new mechanism of lift enhancement, prolonging the benefit of the delayed stall for the entire stroke. Recently, Birch and Dickinson(2001) measured the flow field of a model fruit fly wing in flapping motion, which had a much smaller Reynolds number (Re≈70). They also showed that the LEV did not shed during the translatory phase of an up- or downstroke.

Dickinson et al. (1999)conducted force measurements on flapping robotic fruit fly wings and showed that, in the case of advanced rotation, in addition to the large lift force during the translatory phase of a stroke, large lift peaks occurred at the beginning and near the end of the stroke. (In the case of symmetrical rotation, the lift peak at the beginning became smaller and was followed by a dip, and the lift peak at the end of the stroke also became smaller; in the case of delayed rotation, no lift peak appeared and a large dip occurred at the beginning of the stroke.) Recently, Sun and Tang(2002) simulated the flow around a model fruit fly wing using the method of computational fluid dynamics and confirmed the results of Dickinson et al.(1999). Using simultaneously obtained forces and flow structures, they showed that, in the case of advanced rotation, the large lift peak at the beginning of the stroke was due to the fast acceleration of the wing and that the large lift peak near the end of the stroke was due to the fast pitching-up rotation of the wing. They also explained the behaviour of forces in the cases of symmetrical and delayed rotation. As a result of these studies(Dickinson and Götz, 1993; Ellington et al., 1996; Dickinson et al., 1999; Birch and Dickinson, 2001; Sun and Tang, 2002), we are better able to understand how the fruit fly produces large lift forces. Although the above results were mainly derived from studies on fruit flies, it is quite possible that they are applicable to other insects that employ similar kinematics.

The power requirements for generating lift through the unsteady mechanisms described above cannot be calculated using methods based on steady-state theory. In the rapid acceleration and fast pitching-up rotation mechanisms,virtual-mass force and force due to the generation of the `starting' vortex exist, and they cannot be included in the power calculation using steady-state theory. In the delayed stall mechanism, the dynamic-stall vortex is carried by the wing in its translation, and the drag of the wing, and hence the power required, must be different from that estimated using steady-state theory. It is of great interest to determine the power required for generating lift through the unsteady mechanisms described above. Moreover, when the wing generates a large lift force through these unsteady mechanisms, a large drag force is also generated. For the fruit fly wing, the drag is significnatly larger than the lift, as can be seen from the experimental data(Dickinson et al., 1999; Sane and Dickinson, 2001) and the computational results (Sun and Tang,2002). From the computational results of Sun and Tang(2002), it is estimated that the mean drag coefficient over an up- or downstroke is more than 35% greater than the mean lift coefficient. It is, therefore, of interest to determine whether these unsteady lift mechanisms are realistic from the energetics point of view.

Here, we investigate these problems for hovering flight in Drosophila virilis using computational fluid dynamics. In this method, the pressure and velocity fields around the flapping wing are obtained by solving the Navier-Stokes equations numerically, and the lift and torques due to the aerodynamic forces are calculated on the basis of the flow pressure and velocities. The inertial torques due to the acceleration and rotation of the wing mass can be calculated analytically. The mechanical power required for the flight may be calculated from the aerodynamic and inertial torques. The motion of the flapping wing and the reference frames are illustrated in Fig. 1.

The wing considered in the present study is the same as that used in our previous work on the unsteady lift mechanism(Sun and Tang, 2002). The planform of the wing is similar to that of a fruit fly wing (see Fig. 2). The wing section is an ellipse of 12% thickness (the radius of the leading and trailing edges is 0.2%of the chord length of the aerofoil). The azimuthal rotation of the wing about the Z axis (see Fig. 1A) is called translation, and the pitching rotation of the wing near the end of a stroke and at the beginning of the following stroke is called rotation or flip. The speed at the span location r₂is called the translational speed of the wing, where r₂ is the radius of the second moment of wing area, defined by r₂=(∫_sr²dS/S)^1/2,where r is the radial position along the wing and S is the wing area. The wing length R is 2.76c, and r₂ is 1.6c or 0.58R, where c is the mean chord length.

In the flapping motion described above, the mean flapping velocity U, velocity at midstroke U_m, angle of attack at midstroke α_m, deceleration/acceleration durationΔτ _t, wing rotation duration Δτ_r,period of the flapping cycle τ_c and flip timingτ _r must be given. These parameters will be determined using available flight data, together with the force balance condition of the flight.

In the flapping motion considered in the present paper, the wing conducts translational motion (azimuthal rotation) and pitching rotation. To calculate the flow around a body performing unsteady motion (such as the present flapping wing), one approach is to write and solve the governing equations in a body-fixed, non-inertial reference frame with inertial force terms added to the equations. Another approach is to write and solve the governing equations in an inertial reference frame. By using a time-dependent coordinate transformation and the relationship between the inertial and non-inertial reference frames, a body-conforming computational grid in the inertial reference frame (which varies with time) can be obtained from a body-conforming grid in the body-fixed, non-inertial frame, which needs to be generated only once. This approach has some advantages. It does not need special treatment on the far-field boundary conditions and, moreover, since no extra terms are introduced into the equations, existing numerical methods can be applied directly to the solutions of the equations. This approach is employed here.

The flow equations are solved using the algorithm developed by Rogers and Kwak (1990) and Rogers et al.(1991). The algorithm is based on the method of artificial compressibility, which introduces a pseudotime derivative of pressure into the continuity equation. Time accuracy in the numerical solutions is achieved by subiterating in pseudotime for each physical time step. The algorithm uses a third-order flux-difference splitting technique for the convective terms and a second-order central difference for the viscous terms. The time derivatives in the momentum equation are differenced using a second-order, three-point, backward-difference formula. The algorithm is implicit and has second-order spatial and time accuracy. For details of the algorithm, see Rogers and Kwak(1990) and Rogers et al.(1991). A body-conforming grid was generated using a Poisson solver based on the work of Hilgenstock(1988). The grid topology used in this work was an O—H grid topology. A portion of the grid used for the wing is shown in Fig. 2.

Data for free hovering flight of the fruit fly Drosophila virilisSturtevant were taken from Weis-Fogh(1973), which were derived from Vogel's studies of tethered flight(Vogel, 1966). Insect weight was 1.96×10^-5 N, wing mass was 2.4×10^-6 g(for one wing), wing length R was 0.3 cm, the area of both wings S_t was 0.058 cm², mean chord length cwas 0.097 cm, stroke amplitude Φ was 2.62rad and stroke frequency n was 240 s^-1.

From the above data, the mean translational velocity of the wing U(the reference velocity), the Reynolds number Re, the non-dimensional period of the flapping cycle τ_c and mean lift coefficient required for supporting the insect's weight C̄_L,w were calculated as follows: U=2ϕnr₂=218.7 cm s^-1; Re=cU/v=147 (v=0.144 cm² s^-1);τ _c=(1/n)/(U/c)=8.42; C̄_L,w=1.96×10^-5/0.5ρU²S_t=1.15(ρ=1.23×10^-3 g cm^-3). Note that, in our previous work (Sun and Tang,2002), a smaller C̄_L,w (approximately 0.8)was obtained for the same insect under the same flight conditions. This is because a larger reference velocity, the velocity at midstroke, was used there. The moments and product of inertia were calculated by assuming that the wing mass was uniformly distributed over the wing planform, and the results were as follows: I_xx=0.721×10^-7 g cm²; I_yy=0.069×10^-7 g cm²; I_zz=0.790×10^-7 g cm²; I_xy=0.148×10^-7 g cm².

The code used here, which is based on the flow-computational method outlined above, was developed by Lan and Sun(2001a). It was verified by the analytical solutions of the boundary layer flow on a flat plate(Lan and Sun, 2001a) and the flow at the beginning of a suddenly started aerofoil(Lan and Sun, 2001b) and tested by comparing the calculated and measured steady-state pressure distributions on a wing (Lan and Sun,2001a). To establish further the validity of the code in calculating the unsteady aerodynamic force on flapping wings, we present below a comparison between our calculated results and the experimental data of Dickinson et al. (1999). In our previous work on the fruit fly wing in flapping motion(Sun and Tang, 2002),calculated results using this code were compared with the experimental data,but the wing aspect ratio was not the same as that used in the experiments and only one case was compared.

Measured unsteady lift in Dickinson et al.(1999) was presented in dimensional form. For comparison, we need to convert it into the lift coefficient. In their experiment, the fluid density ρ was 0.88×10³ kg m^-3, the model wing length Rwas 0.25 m and the wing area S was 0.0167 m². The speed at the wing tip during the constant-speed translational phase of a stroke, given in Fig. 3D of Dickinson et al.(1999), was 0.235 m s^-1 and, therefore, the reference speed (mean speed at r₂=0.58R) was calculated to be U=0.118 m s^-1. From the above data,0.5ρU²S=0.102 N. Using the definition of C_L (equation 10), the lift in Fig. 3A of Dickinson et al.(1999) can be converted to C_L (Fig. 4)and compared with the calculated C_L for the cases of advanced rotation (Fig. 4A),symmetrical rotation (Fig. 4B)and delayed rotation (Fig. 4C). The aspect ratio of the wing in the experiment was calculated as R²/S=3.74, and a wing of the same aspect ratio was used in the calculation. The magnitude and trends with variation over time of the calculated C_L are in reasonably good agreement with the measured values.

In the above calculation, the computational grid had dimensions of 93×109×71 in the normal direction, around the wing section and in the spanwise direction, respectively. The normal grid spacing at the wall was 0.002. The outer boundary was set at 10 chord lengths from the wing. The time step was 0.02. Detailed study of the numerical variables such as grid size,domain size, time step, etc., was conducted in our previous work on the unsteady lift mechanism of a flapping fruit fly wing(Sun and Tang, 2002), where it was shown that the above values for the numerical variables were appropriate for the flow calculation. Therefore, in the following calculation, the same set of numerical variables was used.

Since we wanted to study the power requirements for balanced flight, we first investigated the force balance. For the flapping motion considered in the present study, the mean drag on the wing over each flapping cycle was zero, and the horizontal force was balanced. Therefore, we needed only to consider under what conditions the weight of the insect was balanced by the mean lift.

As noted above, the kinematic parameters required to describe the wing motion are U, U_m⁺, τ_c,Δτ _t, Δτ_r, τ_r andα _m. Of these, U and τ_c were determined above using the flight data given by Weis-Fogh(1973). Ennos(1989) made observations of the free forward flight of Drosophila melanogaster (the flight was approximately balanced). His data (his Figs 6D, 7A) showed that symmetrical rotation was employed by the insect. Data on the hovering flight of craneflies, hoverflies and droneflies also showed that symmetrical rotation was employed by these insects (see Figs 8, 9 and 12, respectively, of Ellington, 1984a). Therefore,it was assumed here that symmetrical rotation was employed in hovering flight in Drosophila virilis; as a result, τ_r was determined. From the data of Ennos (1989)and Ellington (1984a), the deceleration/ acceleration duration Δτ_t was estimated to be approximately 0.2τ_c. We assumed thatΔτ _t=0.18τ_c (this value was used in previous studies on unsteady force mechanism of the fruit fly wing; Dickinson et al., 1999; Sun and Tang, 2002). UsingΔτ _t and U, U_m⁺ could be determined.

Dickinson et al. (1999) and Sun and Tang (2002) usedΔτ _r≈0.36τ_c. In the present study, we first assumed Δτ_r≈0.36τ_c and then investigated the effects of varying Δτ_r. At this point,all the kinematic parameters had been determined except α_m,which was determined using the force balance condition.

The calculated lift coefficients versus time for three values ofα _m are shown in Fig. 5. The mean lift coefficient

As shown above, at α_m=35°, the insect produced enough lift to support its weight. In the following, we calculated the power required to produce this lift and investigated the mechanical power output of insect flight muscle and its mechanochemical efficiency.

As expressed in equation 20, the aerodynamic power consists of two components, one due to the aerodynamic torque for translation and the other due to the aerodynamic torque for rotation. The coefficients of these two torques, C_Q,a,t and C_Q,a,r, are shown in Fig. 7B. C_Q,a,t is much larger than C_Q,a,r. The C_Q,a,t curve is similar in shape to the C_D curve shown in Fig. 5C for obvious reasons.

One might expect that, during the deceleration of the wing near the end of a stroke, C_Q,a,t would change sign because of the wing being `pushed' by the flow behind it. But as seen from Fig. 7B (e.g. during the downstroke), C<sub>Q,a,t</sub> becomes negative only when the deceleration is almost finished because, while decelerating, the wing rotates around an axis that is near its leading edge. Therefore, a large part of the wing is effectively not in deceleration and does not `brake' the pushing flow.

The coefficients of the inertial torques for translation(C_Q,i,t) and for rotation (C_Q,i,r) are shown in Fig. 7C. The inertial torques are approximately zero in the middle of a stroke, when the translational and rotational accelerations are zero. At the beginning and near the end of the stroke, the inertial torque for translation has almost the same magnitude as its aerodynamic counterpart. Similar to the case of the aerodynamic torques, the inertial torque for translation is much larger than the inertial torque for rotation.

At the beginning of a stroke, the sign of C_Q,i,r is opposite to that of

From the above results for the aerodynamic and inertial torque coefficients, the power coefficients can be computed using equations 41-43. The coefficients of power for translation (C_P,t) and rotation (C_P,r) are plotted against time in Fig. 8. C_P,t is positive for the majority of a stroke and becomes negative only for a very short period close to the end of the stroke. C_P,r is negative at the beginning and near the end of a stroke and is approximately zero in the middle of the stroke. Throughout a stroke, the magnitude of C_P,t is much larger than that of C_P,r. Two large positive peaks in C_P,tappear during a stroke. One occurs during the rapid acceleration phase of the stroke as a result of the larger aerodynamic and inertial torques occurring there. The other is in the fast pitching-up rotation phase of the stroke and is due to the large aerodynamic torque there.

Integrating C_P,t over the part of a cycle where it is positive gives the coefficient of positive work for translation, which is represented by C_W,t⁺. Integrating C_P,t over the part of the cycle where it is negative gives the coefficient of `negative' work for translation; this is represented by C_W,t^-. Similar integration of C_P,r gives the coefficients of the positive and negative work for rotation; they are denoted by C_W,r⁺and C_W,r^-, respectively. The results of the integration are: C_W,t⁺=15.96, C_W,t^-=-1.00, C_W,r⁺=0.56 and C_W,r^-=-2.30.

Using C_W,t⁺ and C_W,r⁺ calculated above, C_Wwas calculated using equation 45 to be 16.52. The specific power P^* was then calculated using equation 44: P^*=36.7 W kg^-1.

In the flight considered above, symmetrical rotation was employed. Dickinson et al. (1999) showed that the timing of the wing rotation had significant effects on the lift and drag of the wing. It is of interest to see how the lift and the power required change when the timing of wing rotation is varied.

Fig. 9 shows the calculated lift and drag coefficients for the cases of advanced rotation and delayed rotation (results for the case of symmetrical rotation are included for comparison). The value of τ_r used can be read from Fig. 9A. The case of advanced rotation has a larger C_L and C_D than the case of symmetrical rotation, and the case of delayed rotation has a much smaller C_L and a slightly larger C_Dthan the case of symmetrical rotation. An explanation for the above force behaviours was given by Dickinson et al.(1999) and Sun and Tang(2002). The mean lift coefficient

The aerodynamic and inertial torque coefficients for the advanced rotation and delayed rotation cases are shown in Figs 10 and 11, respectively. Similar to the symmetrical rotation case, the C_Q,a,t curve for the case of advanced rotation (or delayed rotation) looks like the corresponding C_D curve. For the advanced rotation case, C_Q,a,t is much larger than that of the symmetrical rotation case, especially from the middle to the end of the stroke (compare Fig. 10B with Fig. 7B). C_Q,i,t is the same as that of the symmetrical rotation case, because the translational acceleration is the same for the two cases(compare Fig. 10C with Fig. 7C). For the delayed rotation case, C_Q,a,t is larger than that of the symmetrical rotation case in the early part of a stroke (compare Fig. 11B with Fig. 7B). C_Q,i,t is the same as that of the symmetrical rotation case for the same reason as above. Similar to the symmetrical rotation case,for both the advanced and delayed rotation cases, C_Q,a,rand C_Q,i,r are much smaller than C_Q,a,t and C_Q,i,t, respectively.

The non-dimensional power coefficients for the cases of advanced rotation and delayed rotation are shown in Fig. 12 (the results of symmetrical rotation, taken from Fig. 8, are included for comparison). C_P,r is much smaller than C_P,t because, as shown in Figs 10 and 11, C_Q,a,rand C_Q,i,r were much smaller than C_Q,a,t and C_Q,i,t, respectively. C_P,t behaves approximately the same as C_Q,a,t. The most striking feature of Fig. 12 is that C_P,t for advanced rotation is much larger than that for symmetrical rotation from the middle to near the end of a stroke.

Integrating the power for the cases of advanced rotation and delayed rotation in the same way as above for the case of symmetrical rotation, the corresponding values of C_W,t⁺, C_W,t-, C_W,r⁺ and C_W,r^- were obtained, from which the work coefficient per cycle, C_W, was calculated. The results are given in Table 1 (results for symmetrical rotation are included for comparison). For the advanced rotation case, C_W is approximately 80 % larger than for symmetrical rotation case. As noted above,

For reference, we calculated another case in which advanced rotation timing was employed in balanced flight but α_m was decreased to 22° so that the mean lift was equal to the weight(Table 1). In this case, C_W was 30.10, which is approximately 82 % larger than for the case employing symmetrical rotation, showing clearly that advanced rotation is very energy-demanding.

In the above analyses, the duration of wing rotation (or flip duration) was taken as Δτ_r=0.36τ_c. Below, the effects of changing the flip duration were investigated. Fig. 13 shows the lift and drag coefficients of the wing for two shorter flip durationsΔτ _r=0.24τ_c andΔτ _r=0.19τ_c, with the above results forΔτ _r=0.36τ_c included for comparison. If the flip duration is varied while other parameters are kept unchanged, the mean lift coefficient will change. To maintain the balance between mean lift and insect weight, α_m was therefore adjusted. ForΔτ _r=0.24τ_c, α_m was changed to 36.5° to give a

From Fig. 13C, it can be seen that, when Δτ_r is decreased, the C_D peak at the beginning of a stroke becomes much smaller and the C_D peak near the end of the stroke is delayed and becomes slightly smaller. A smaller C_D at the beginning of the stroke means reduced aerodynamic power there. Since the wing decelerates near the end of the stroke, delaying the C_D peak at this point means that the peak would occur when the wing has a lower velocity,resulting in reduced aerodynamic power. The power coefficients are shown in Fig. 14; at the beginning and at the end of a stroke, C_P,t is smaller for smallerΔτ _r.

By integrating the power coefficients in Fig. 14, C_W,t⁺, C_W,t^-, C_W,r⁺ and C_W,r^-were obtained (Table 2). C_W was computed using equation 45, and the results are also shown in Table 2. WhenΔτ _r is decreased to 0.24τ_c, C_W is 12.96, much smaller than forΔτ _r=0.36τ_c. When Δτ_ris further decreased to 0.19τ_c, C_W was slightly greater (13.06). This is because when Δτ_r is decreased to 0.19τ_c, C_W,t⁺ also decreases; however, C_W,r⁺ increases (possibly due to the wing rotation becoming very fast).

It is very interesting to look at the drag on the wing. In the above case(Δτ_r=0.24τ_c,α _m=36.5°), the mean drag coefficient

The above results showed that, for a duration of wing rotationΔτ _r=0.19τ_c andΔτ _r=0.24τ_c, the power expenditure for hovering flight in Drosophila virilis was relatively small compared with that for larger values of Δτ_r. The corresponding non-dimensional mean rotational velocities are approximately 1 (maximum

The calculated midstroke angle of attack α_m is approximately 37° (see Table 2; α_m=36.5° and α_m=38.5°for Δτ_r=0.24τ_c andΔτ _r=0.19τ_c, respectively). Vogel(1967) measured the angle of attack for tethered Drosophila virilis flying in still air;α _m was approximately 45°. Our calculated value is smaller than this value, which is reasonable since the calculated value is for free and balanced flight whereas the measured value was for tethered flight in which the insect could use a larger angle of attack. Ellington(1984a) observed many small insects in hovering flight, including the fruit fly, and found that the angle of attack employed was approximately 35°. The predicted value thus is in good agreement with observations.

The calculated body-mass-specific power P^* and muscle efficiency η were 28.7 W kg^-1 and 17 %, respectively. Lehmann and Dickinson (1997) studied the muscle efficiency of the fruit fly Drosophila melanogaster by simultaneously measuring the metabolic rate and the flight kinematics. Using the measured stroke amplitude and frequency, they estimated the mean specific power using a quasi-steady aerodynamics method. Their estimate of P^* for hovering flight was 17.9 W kg^-1, only approximately half the value calculated using the present unsteady flow simulation method. Their measured metabolic rate was approximately 199 W kg^-1. As a result, they obtained a muscle efficiency of approximately 9 %, approximately half that obtained in the present study. In their recent work on unsteady force measurements on a model fruit fly wing,Sane and Dickinson (2001)showed that the drag on the wing was much larger than the quasi-steady estimate of Lehmann and Dickinson(1997). On the basis of the measured drag, they suggested that the previous value of muscle efficiency presented by Lehmann and Dickinson(1997) should be adjusted to approximately 20 %. This is similar to the value calculated in the present study.

The calculated results show that, for the advanced rotation and delayed rotation cases, the energy expended for a given mean lift is much larger than that in the case of symmetrical rotation. On the basis of these results,symmetrical rotation should be employed by the insect for balanced,long-duration flight and advanced rotation and delayed rotation should be employed for manoeuvring. This agrees with observations on balanced flight(Ennos, 1989) and manoeuvring(Dickinson et al., 1993) in the fruit fly Drosophila melanogaster.

List of symbols
 
• c

  mean chord length

 
• C_D

  drag coefficient

 
• \(C_{\mathrm{D}}\)

  mean drag coefficient (over an up- or downstroke)

 
• C_L

  lift coefficient

 
• \(C_{\mathrm{L}}\)

  mean lift coefficient

 
• C̄ _L,w

  mean lift coefficient for supporting the insect's weight

 
• C_P

  power coefficient

 
• C_P,a

  coefficient of aerodynamic power

 
• C_P,i

  coefficient of inertial power

 
• C_P,r

  coefficient of power for rotation

 
• C_P,t

  coefficient of power for translation

 
• C_Q,a,r

  coefficient of aerodynamic torque for rotation

 
• C_Q,a,t

  coefficient of aerodynamic torque for translation

 
• C_Q,i,r

  coefficient of inertial torque for rotation

 
• C_Q,i,t

  coefficient of inertial torque for translation

 
• C_W

  coefficient of work per cycle

 
• C_W,r⁺

  coefficient of positive work for rotation

 
• C_W,r^-

  coefficient of negative work for rotation

 
• C_W,t⁺

  coefficient of positive work for translation

 
• C_W,t^-

  coefficient of negative work for translation

 
• D

  drag

 
• dm_w

  mass element of the wing

 
• F

  aerodynamic force per unit wing area

 
• f_x, f_y, f_z

  x, y and z components of F,respectively

 
• H

  angular momentum of a wing

 
• i, j, k

  unit vectors in the x, y and z directions,respectively

 
• I_xx, I_yy, I_zz

  moments of inertia of the wing about the x, y and zaxes, respectively

 
• I_xy, I_yz, I_xz

  products of inertia of the wing

 
• L

  lift

 
• m

  mass of the insect

 
• m_w

  wing mass of the insect

 
• M_a

  aerodynamic moment

 
• M_i

  inertial moment

 
• m_i^x, m_i^y, m_i^z

  x, y and z components of M_i,respectively

 
• n

  wingbeat frequency

 
• O, o

  origins of the inertial and non-inertial frames of reference,repectively

 
• p

  non-dimensional fluid pressure

 
• P

  mechanical power

 
• P_a

  aerodynamic power

 
• P_i

  inertial power

 
• P^*

  body-mass-specific power

 
• P_m^*

  muscle-mass-specific power

 
• Q_a,r

  aerodynamic torque for rotation

 
• Q_a,t

  aerodynamic torque for translation

 
• Q_i,r

  inertial torque for rotation

 
• Q_i,t

  inertial torque for translation

 
• r

  radial position along wing length

 
• r₂

  radius of the second moment of wing area

 
• r

  vector distance between point o and an element on the wing

 
• R

  wing length

 
• Re

  Reynolds number

 
• R_m

  body-mass-specific metabolic rate

 
• S

  area of one wing

 
• S_t

  area of a wing pair

 
• t

  time

 
• u, v, w

  three components of non-dimensional fluid velocity

 
• u_t

  translational velocity of the wing

 
• u_t⁺

  non-dimensional translational velocity of the wing

 
• U

  reference velocity (u_t averaged over a stroke)

 
• U_m

  midstroke translational velocity of a wing (or maximum of u_t)

 
• U_m⁺

  maximum of u_t⁺

 
• X, Y, Z

  coordinates in the inertial frame of reference

 
• x, y, z

  coordinates in the non-inertial frame of reference

 
• α

  angle of attack

 
• αm

  midstroke angle of attack

 
• \({\dot{{\alpha}}}\)

  angular velocity of pitching rotation

 
• \({\dot{{\alpha}}}^{+}\)

  non-dimensional angular velocity of pitching rotation

 
• \({\dot{{\alpha}}}_{0}^{+}\)

  a constant

 
• \({\ddot{{\alpha}}}\)

  angular acceleration of pitching rotation

 
• \({\ddot{{\alpha}}}^{+}\)

  non-dimensional angular acceleration of pitching rotation

 
• Δτt

  duration of deceleration/acceleration around stroke reversal(non-dimensional)

 
• Δτr

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

 
• η

  muscle efficiency

 
• ν

  kinematic viscosity

 
• ϕ

  azimuthal or positional angle

 
• Φ

  stroke amplitude

 
• \({\dot{{\phi}}}\)

  angular velocity of azimuthal rotation

 
• \({\dot{{\phi}}}^{+}\)

  non-dimensional angular velocity of azimuthal rotation

 
• \({\ddot{{\phi}}}\)

  angular acceleration of azimuthal rotation

 
• \({\ddot{{\phi}}}^{+}\)

  non-dimensional angular acceleration of azimuthal rotation

 
• ρ

  density of fluid

 
• τ

  non-dimensional time

 
• τc

  period of one flapping cycle (non-dimensional)

 
• τ1

  time when translational deceleration starts (non-dimensional)

 
• τr

  time when pitching rotation starts (non-dimensional)

 
• τt

  time when stroke reversal starts (non-dimensional)

 
• τs

  period of one stroke (non-dimensional)

 
• τ0

  time when a stroke starts (non-dimensional)

 
• Ω

  total angular velocity vector of the wing

We would like to thank the two referees for their helpful comments on this manuscript. This research was supported by the National Natural Science Foundation of China.