The lift and power requirements for hovering flight in Drosophila virilis were studied using the method of computational fluid dynamics. The Navier-Stokes equations were solved numerically. The solution provided the flow velocity and pressure fields, from which the unsteady aerodynamic forces and moments were obtained. The inertial torques due to the acceleration of the wing mass were computed analytically. On the basis of the aerodynamic forces and moments and the inertial torques, the lift and power requirements for hovering flight were obtained.
For the fruit fly Drosophila virilis in hovering flight (with symmetrical rotation), a midstroke angle of attack of approximately 37°was needed for the mean lift to balance the insect weight, which agreed with observations. The mean drag on the wings over an up- or downstroke was approximately 1.27 times the mean lift or insect weight (i.e. the wings of this tiny insect must overcome a drag that is approximately 27 % larger than its weight to produce a lift equal to its weight). The body-mass-specific power was 28.7 W kg-1, the muscle-mass-specific power was 95.7 W kg-1 and the muscle efficiency was 17 %.
With advanced rotation, larger lift was produced than with symmetrical rotation, but it was more energy-demanding, i.e. the power required per unit lift was much larger. With delayed rotation, much less lift was produced than with symmetrical rotation at almost the same power expenditure; again, the power required per unit lift was much larger. On the basis of the calculated results for power expenditure, symmetrical rotation should be used for balanced, long-duration flight and advanced rotation and delayed rotation should be used for flight control and manoeuvring. This agrees with observations.
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.
Materials and methods
The wing and its kinematics
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 Um, angle of attack at midstroke αm, deceleration/acceleration durationΔτ t, wing rotation duration Δτr,period of the flapping cycle τc 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.
The Navier—Stokes equations and the computational method
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.
Description of the coordinate systems
Evaluation of the aerodynamic forces
Evaluation of the aerodynamic torque and power
Evaluations of the inertial torques and power
Evaluation of the total mechanical power
Data for hovering flight in Drosophila virilis
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 St 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 τc and mean lift coefficient required for supporting the insect's weight C̄L,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; C̄L,w=1.96×10-5/0.5ρU2St=1.15(ρ=1.23×10-3 g cm-3). Note that, in our previous work (Sun and Tang,2002), a smaller C̄L,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: Ixx=0.721×10-7 g cm2; Iyy=0.069×10-7 g cm2; Izz=0.790×10-7 g cm2; Ixy=0.148×10-7 g cm2.
Results and discussion
Test of the flow solver
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 CL (equation 10), the lift in Fig. 3A of Dickinson et al.(1999) can be converted to CL (Fig. 4)and compared with the calculated CL 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 CL 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.
Force balance in hovering flight
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, Um+, τc,Δτ t, Δτr, τr andα m. Of these, U and τc 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, τr was determined. From the data of Ennos (1989)and Ellington (1984a), the deceleration/ acceleration duration Δτt was estimated to be approximately 0.2τc. We assumed thatΔτ t=0.18τc (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, Um+ could be determined.
Dickinson et al. (1999) and Sun and Tang (2002) usedΔτ r≈0.36τc. In the present study, we first assumed Δτr≈0.36τc and then investigated the effects of varying Δτr. At this point,all the kinematic parameters had been determined except αm,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 αm=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, CQ,a,t and CQ,a,r, are shown in Fig. 7B. CQ,a,t is much larger than CQ,a,r. The CQ,a,t curve is similar in shape to the CD curve shown in Fig. 5C for obvious reasons.
One might expect that, during the deceleration of the wing near the end of a stroke, CQ,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), CQ,a,t 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(CQ,i,t) and for rotation (CQ,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 CQ,i,r is opposite to that of
Power and work
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 (CP,t) and rotation (CP,r) are plotted against time in Fig. 8. CP,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. CP,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 CP,t is much larger than that of CP,r. Two large positive peaks in CP,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 CP,t over the part of a cycle where it is positive gives the coefficient of positive work for translation, which is represented by CW,t+. Integrating CP,t over the part of the cycle where it is negative gives the coefficient of `negative' work for translation; this is represented by CW,t-. Similar integration of CP,r gives the coefficients of the positive and negative work for rotation; they are denoted by CW,r+and CW,r-, respectively. The results of the integration are: CW,t+=15.96, CW,t-=-1.00, CW,r+=0.56 and CW,r-=-2.30.
Using CW,t+ and CW,r+ calculated above, CWwas calculated using equation 45 to be 16.52. The specific power P* was then calculated using equation 44: P*=36.7 W kg-1.
Effects of the timing of wing rotation on lift and power
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 τr used can be read from Fig. 9A. The case of advanced rotation has a larger CL and CD than the case of symmetrical rotation, and the case of delayed rotation has a much smaller CL and a slightly larger CDthan 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 CQ,a,t curve for the case of advanced rotation (or delayed rotation) looks like the corresponding CD curve. For the advanced rotation case, CQ,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). CQ,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, CQ,a,t is larger than that of the symmetrical rotation case in the early part of a stroke (compare Fig. 11B with Fig. 7B). CQ,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, CQ,a,rand CQ,i,r are much smaller than CQ,a,t and CQ,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). CP,r is much smaller than CP,t because, as shown in Figs 10 and 11, CQ,a,rand CQ,i,r were much smaller than CQ,a,t and CQ,i,t, respectively. CP,t behaves approximately the same as CQ,a,t. The most striking feature of Fig. 12 is that CP,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 CW,t+, CW,t-, CW,r+ and CW,r- were obtained, from which the work coefficient per cycle, CW, was calculated. The results are given in Table 1 (results for symmetrical rotation are included for comparison). For the advanced rotation case, CW is approximately 80 % larger than for symmetrical rotation case. As noted above,
|αm (degrees) .||Rotation timing .|
|CW,t+ .||CW,t- .||CW,r+ .||CW,r- .||CW .|
|αm (degrees) .||Rotation timing .|
|CW,t+ .||CW,t- .||CW,r+ .||CW,r- .||CW .|
For reference, we calculated another case in which advanced rotation timing was employed in balanced flight but αm was decreased to 22° so that the mean lift was equal to the weight(Table 1). In this case, CW was 30.10, which is approximately 82 % larger than for the case employing symmetrical rotation, showing clearly that advanced rotation is very energy-demanding.
Effects of flip duration
In the above analyses, the duration of wing rotation (or flip duration) was taken as Δτr=0.36τc. 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=0.24τc andΔτ r=0.19τc, with the above results forΔτ r=0.36τc 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, αm was therefore adjusted. ForΔτ r=0.24τc, αm was changed to 36.5° to give a
From Fig. 13C, it can be seen that, when Δτr is decreased, the CD peak at the beginning of a stroke becomes much smaller and the CD peak near the end of the stroke is delayed and becomes slightly smaller. A smaller CD at the beginning of the stroke means reduced aerodynamic power there. Since the wing decelerates near the end of the stroke, delaying the CD 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, CP,t is smaller for smallerΔτ r.
By integrating the power coefficients in Fig. 14, CW,t+, CW,t-, CW,r+ and CW,r-were obtained (Table 2). CW was computed using equation 45, and the results are also shown in Table 2. WhenΔτ r is decreased to 0.24τc, CW is 12.96, much smaller than forΔτ r=0.36τc. When Δτris further decreased to 0.19τc, CW was slightly greater (13.06). This is because when Δτr is decreased to 0.19τc, CW,t+ also decreases; however, CW,r+ increases (possibly due to the wing rotation becoming very fast).
|Δτr/τc .||αm (degrees) .|
|CW,t+ .||CW,t- .||CW,r+ .||CW,r- .||CW .|
|Δτr/τc .||αm (degrees) .|
|CW,t+ .||CW,t- .||CW,r+ .||CW,r- .||CW .|
It is very interesting to look at the drag on the wing. In the above case(Δτr=0.24τc,α m=36.5°), the mean drag coefficient
Comparison between the calculated results and previous data
The above results showed that, for a duration of wing rotationΔτ r=0.19τc andΔτ r=0.24τc, the power expenditure for hovering flight in Drosophila virilis was relatively small compared with that for larger values of Δτr. The corresponding non-dimensional mean rotational velocities are approximately 1 (maximum
The calculated midstroke angle of attack αm is approximately 37° (see Table 2; αm=36.5° and αm=38.5°for Δτr=0.24τc andΔτ r=0.19τc, 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.
mean chord length
mean drag coefficient (over an up- or downstroke)
mean lift coefficient
- C̄ L,w
mean lift coefficient for supporting the insect's weight
coefficient of aerodynamic power
coefficient of inertial power
coefficient of power for rotation
coefficient of power for translation
coefficient of aerodynamic torque for rotation
coefficient of aerodynamic torque for translation
coefficient of inertial torque for rotation
coefficient of inertial torque for translation
coefficient of work per cycle
coefficient of positive work for rotation
coefficient of negative work for rotation
coefficient of positive work for translation
coefficient of negative work for translation
mass element of the wing
aerodynamic force per unit wing area
- fx, fy, fz
x, y and z components of F,respectively
angular momentum of a wing
- i, j, k
unit vectors in the x, y and z directions,respectively
- Ixx, Iyy, Izz
moments of inertia of the wing about the x, y and zaxes, respectively
- Ixy, Iyz, Ixz
products of inertia of the wing
mass of the insect
wing mass of the insect
- mix, miy, miz
x, y and z components of Mi,respectively
- O, o
origins of the inertial and non-inertial frames of reference,repectively
non-dimensional fluid pressure
aerodynamic torque for rotation
aerodynamic torque for translation
inertial torque for rotation
inertial torque for translation
radial position along wing length
radius of the second moment of wing area
vector distance between point o and an element on the wing
body-mass-specific metabolic rate
area of one wing
area of a wing pair
- u, v, w
three components of non-dimensional fluid velocity
translational velocity of the wing
non-dimensional translational velocity of the wing
reference velocity (ut averaged over a stroke)
midstroke translational velocity of a wing (or maximum of ut)
maximum of ut+
- X, Y, Z
coordinates in the inertial frame of reference
- x, y, z
coordinates in the non-inertial frame of reference
angle of attack
midstroke angle of attack
angular velocity of pitching rotation
non-dimensional angular velocity of pitching rotation
angular acceleration of pitching rotation
non-dimensional angular acceleration of pitching rotation
duration of deceleration/acceleration around stroke reversal(non-dimensional)
duration of wing rotation or flip duration (non-dimensional)
azimuthal or positional angle
angular velocity of azimuthal rotation
non-dimensional angular velocity of azimuthal rotation
angular acceleration of azimuthal rotation
non-dimensional angular acceleration of azimuthal rotation
density of fluid
period of one flapping cycle (non-dimensional)
time when translational deceleration starts (non-dimensional)
time when pitching rotation starts (non-dimensional)
time when stroke reversal starts (non-dimensional)
period of one stroke (non-dimensional)
time when a stroke starts (non-dimensional)
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.