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The Journal of Experimental Biology 205, 2413-2427 (2002)
© 2002 The Company of Biologists Limited

Lift and power requirements of hovering flight in Drosophila virilis

Mao Sun^* and Jian Tang

Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of China

^* e-mail: sunmao{at}public.fhnet.cn.net

Accepted 22 May 2002

	Summary

TOP Summary Introduction Materials and methods Results and discussion References

 
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.

Key words: fruit fly, Drosophila virilis, lift, power, unsteady aerodynamics, computational fluid dynamics

	Introduction

TOP Summary Introduction Materials and methods Results and discussion References

 
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 sexta during 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 [Re3500; 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 (Re70). 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.

View larger version (15K): [in this window] [in a new window]   Fig. 1. Sketches of the reference frames and wing motion. (A) OXYZ is an inertial frame, with the XY plane in the stroke plane. oxyz is a frame fixed on the wing, with the x axis along the wing chord, and the y axis along the wing span. is the azimuthal angle of the wing, is the angle of attack and R is the wing length. (B) The motion of a section of the wing.

	Materials and methods

TOP Summary Introduction Materials and methods Results and discussion References

 
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 r₂ is 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.

View larger version (117K): [in this window] [in a new window]   Fig. 2. The wing planform and a portion of the body-conforming grid near the wing in the z=0 plane (see Fig. 1A for a definition of this plane).

The flapping motion considered here is an idealized one, which is similar to that considered by Dickinson et al. (1999) and Sun and Tang (2002) in their studies on unsteady lift mechanisms. An up- or downstroke consists of the following three parts, as shown in Fig. 1B: pitching-down rotation and translational acceleration at the beginning of the stroke, translation at constant speed and constant angle of attack during the middle of the stroke, and pitching-up rotation and translational deceleration at the end of the stroke. The translational velocity is denoted by u_t, which takes a constant value of U_m except at the beginning and near the end of a stroke. During the acceleration at the beginning of a stroke, u_t is given by:

(1)

where u_t⁺=U_t/U (U is the mean value of u_t over a stroke and is used as reference velocity and u_t⁺ is the non-dimensional translational velocity of the wing), U_m⁺=U_m/U (U_m⁺ is the maximum of u_t⁺), =tU/c (t is dimensional time and is the non-dimensional time), ₀ is the non-dimensional time at which the stroke starts and ₀+(_t/2) is the time at which the acceleration at the beginning of the stroke finishes. _t is the duration of deceleration/acceleration around stroke reversal. Near the end of the stroke, the wing decelerates from u_t⁺=U_m⁺ to u_t⁺=0 according to:

(2)

where ₁ is the non-dimensional time at which the deceleration starts. The azimuth-rotational speed of the wing is related to u_t. Denoting the azimuthal-rotational speed as , we have .

The angle of attack of the wing is denoted by . It also takes a constant value except at the beginning or near the end of a stroke. The constant value is denoted by _m. Around the stroke reversal, changes with time and the angular velocity is given by:

(3)

where the non-dimensional form is a constant, _r is the time at which the rotation starts and _r is the time interval over which the rotation lasts. In the time interval _r, the wing rotates from =_m to =180 °-_m. Therefore, when _m and _r are specified, can be determined (around the next stroke reversal, the wing would rotate from =180 °-_m to =_m; the sign of the right-hand side of equation 3 should then be reversed). The axis of the pitching rotation is located 0.2c from the leading edge of the wing. _r is termed wing rotation duration (or flip duration), and _r is termed the rotation (or flip) timing, When _r is chosen such that the majority of the wing rotation is conducted near the end of a stroke, it is called advanced rotation; when _r is chosen such that half the wing rotation is conducted near the end of a stroke and half at the beginning of the next stroke, it is called symmetrical rotation; when _r is chosen such that the major part of the wing rotation is delayed to the beginning of the next stroke, it is called delayed rotation.

In the flapping motion described above, the mean flapping velocity U, velocity at midstroke U_m, angle of attack at midstroke _m, deceleration/acceleration duration _t, wing rotation duration _r, period of the flapping cycle _c and flip timing _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
The flow equations and computational method used in the present study are the same as in a recent paper (Sun and Tang, 2002). Therefore, only an outline of the method is given here. The governing equations of the flow are the three-dimensional incompressible unsteady Navier—Stokes equations. Written in the inertial coordinate system OXYZ and non-dimensionalized, they are as follows:

(4)

(5)

(6)

(7)

where u, v and w are three components of the non-dimensional fluid velocity and p is the non-dimensional fluid pressure. In the non-dimensionalization, U, c and c/U are taken as reference velocity, length and time, respectively. Re in equations 5-7 denotes the Reynolds number and is defined as Re=cU/v, where v is the kinematic viscosity of the fluid.

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
In both the flow calculation method outlined above and the force and moment calculations below, two coordinate systems are needed. They are described as follows. One is the inertial coordinate system OXYZ. The origin O is at the root of the wing. The X and Y axes are in the horizontal plane with the X axis positive aft, the Y axis positive starboard and the Z axis positive vertically up (see Fig. 1A). The second is the body-fixed coordinate system oxyz. It has the same origin as the inertial coordinate system, but it rotates with the wing. The x axis is parallel to the wing chord and positive aft, and the y axis is on the pitching-rotation axis of the wing and positive starboard (see Fig. 1A). In terms of the Euler angles and (defined in Fig. 1A), the relationship between these two coordinate systems is given by:

(8)

and

(9)

Evaluation of the aerodynamic forces
Once the Navier—Stokes equations have been numerically solved, the fluid velocity components and pressure at discretized grid points for each time step are available. The aerodynamic force acting on the wing is contributed by the pressure and the viscous stress on the wing surface. Integrating the pressure and viscous stress over the wing surface at a time step gives the total aerodynamic force acting on the wing at the corresponding instant in time. The lift of the wing, L, is the component of the total aerodynamic force perpendicular to the translational velocity and is positive when directed upwards. The drag, D, is the component of the total aerodynamic force parallel to the translational velocity and is positive when directed opposite to the direction of the translational velocity of the downstroke. The lift and drag coefficients, denoted by C_L and C_D, respectively, are defined as follows:

(10)

(11)

where is the fluid density.

Evaluation of the aerodynamic torque and power
The moment around the root of a wing (point o) due to the aerodynamic forces, denoted by -M_a, can be written as follows (assuming that the thickness of the wing is very small):

(12)

where S is the wing surface area, F is the aerodynamic force in unit wing area, r is the distance vector, f_x, f_y and f_z are the three components of F in the oxyz coordinate system, x, y and z are the three components of r, and i, j and k are the unit vectors of the coordinate system oxyz (see Fig. 3). In equation 12, f_y, the component of F along the wing span, is neglected and f_x and f_z can be obtained from the solution of the flow equations. The angular velocity vector of the wing, , has the following three components in the oxyz coordinate system:

(13)

View larger version (7K): [in this window] [in a new window]   Fig. 3. Diagram showing how the moments and product of inertia are computed. o, x, y, coordinates in wing-fixed frame of reference; dmw, mass element of the wing; r, vector distance between point o and a mass element of the wing; i, j, unit vectors in the x and y directions, respectively.

The power required to overcome the aerodynamic moments, called aerodynamic power P_a, can be written as:

(14)

where

(15)

(16)

Q_a,t is the torque around the axis of azimuthal rotation and is due to the aerodynamic drag. It is termed the aerodynamic torque for translation. Q_ar is the torque around the axis of pitching rotation and is due to the aerodynamic pitching moment. It is termed the aerodynamic torque for rotation. The coefficients of the aerodynamic torques are defined as follows:

(17)

(18)

The aerodynamic power coefficient is defined as:

(19)

and can be written as:

(20)

where is the non-dimensional-angular velocity of azimuthal rotation.

Evaluations of the inertial torques and power
The moments and products of inertia of the mass of a wing, with respect to the oxyz coordinate system (see Fig. 3), can be written as follows, assuming that the thickness of the wing is very small:

(21)

(22)

(23)

(24)

(25)

(26)

where dm_w is a mass element of the wing. The angular momentum of the wing H and its three components in the oxyz coordinate system are as follows:

(27)

where i, j and k are unit vectors in the x, y and z directions, respectively. The inertial moment of the wing, M_i, is:

(28)

where (dH/dt)_xyz denotes the time derivative of the vector as observed in the rotating system and m_i^x, m_i^y and m_i^z, the x, y and z components of M_i, respectively, are as follows:

(29)

(30)

(31)

where and are the angular acceleration of the azimuthal and pitching rotations, respectively. In deriving equation 28, I_yyI_zz-I_zz was used. The inertial power of the wing is written as follows:

(32)

where

(33)

(34)

Q_i,t is the inertial torque around the axis of azimuthal rotation and is termed the inertial torque for translation. Q_i,r is the inertial torque around the axis of pitching rotation and is termed the inertial torque for rotation. The coefficients of P_i, Q_i,t and Q_i,r are denoted by C_P,i, C_Q,i,t and C_Q,i,r, respectively, and are defined in the same way as the coefficients of the aerodynamic power and aerodynamic torques. From equation 32, the inertial power coefficient can be written as:

(35)

and the expressions for C_Q,i,t and C_Q,i,r are as follows:

(36)

(37)

where and are the non-dimensional angular acceleration of the azimuthal and pitching rotations, respectively. Using the values of I_xx, I_yy, I_zz and I_xy given in the next section, C_Q,i,t and C_Q,i,r can be written as:

(38)

(39)

From equations 36 and 37, or equations 38 and 39, it can be seen that the translational motion also contributes to the rotational inertial torque and vice versa. Since I_xy is over twice as large as I_yy, the contribution from the translational acceleration to the rotational inertial torque can be larger than that from the rotational acceleration .

Evaluation of the total mechanical power
The total mechanical power of the wing, P, is the power required to overcome the combination of the aerodynamic and inertial torques and can be written as:

(40)

Henceforth, it is termed simply the power. Combining equations 20 and 35, the non-dimensional power coefficient (represented by C_P), can be written as follows:

(41)

where

(42)

(43)

C_P,t is the coefficient of power for translation and C_P,r the coefficient of power for rotation.

Data for hovering flight in Drosophila virilis
Data for free hovering flight of the fruit fly Drosophila virilis Sturtevant were taken from Weis-Fogh (1973), which were derived from Vogel's studies of tethered flight (Vogel, 1966). Insect weight was 1.96x10^-5 N, wing mass was 2.4x10^-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 c was 0.097 cm, stroke amplitude 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 _c and mean lift coefficient required for supporting the insect's weight _L,w were calculated as follows: U=2nr₂=218.7 cm s^-1; Re=cU/v=147 (v=0.144 cm² s^-1); _c=(1/n)/(U/c)=8.42; _L,w=1.96x10^-5/0.5U²S_t=1.15 (=1.23x10^-3 g cm^-3). Note that, in our previous work (Sun and Tang, 2002), a smaller _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=0.721x10^-7 g cm²; I_yy=0.069x10^-7 g cm²; I_zz=0.790x10^-7 g cm²; I_xy=0.148x10^-7 g cm².

	Results and discussion

TOP Summary Introduction Materials and methods Results and discussion References

 
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 was 0.88x10³ kg m^-3, the model wing length R was 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, 0.5U²S=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.

View larger version (23K): [in this window] [in a new window]   Fig. 4. Comparison of the calculated lift coefficient CL with the measured CL. The experimental data are reproduced from Fig. 3 of Dickinson et al. (1999). (A) Advanced rotation. (B) Symmetrical rotation. (C) Delayed rotation. , non-dimensional time; c, non-dimensional period of one flapping cycle.

In the above calculation, the computational grid had dimensions of 93x109x71 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, U_m⁺, _c, _t, _r, _r and _m. Of these, U and _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, _r was determined. From the data of Ennos (1989) and Ellington (1984a), the deceleration/ acceleration duration _t was estimated to be approximately 0.2_c. We assumed that _t=0.18_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). Using _t and U, U_m⁺ could be determined.

View larger version (8K): [in this window] [in a new window]   Fig. 6. Mean lift coefficient versus midstroke angle of attack m (symmetrical rotation, non-dimensional duration of wing rotation r=0.36c, where c is the non-dimensional period of one flapping cycle).

View larger version (20K): [in this window] [in a new window]   Fig. 7. Non-dimensional angular velocity of pitching rotation and azimuthal rotation (A), aerodynamic (B) and inertial (C) torque coefficients for translation (CQ,a,t, and CQ,i,t, respectively) and rotation (CQ,a,r and CQ,i,r, respectively) versus time during one cycle (midstroke angle of attack m=35°, symmetrical rotation, non-dimensional duration of wing rotation r=0.36c). c, non-dimensional period of one flapping cycle; , non-dimensional time.

View larger version (17K): [in this window] [in a new window]   Fig. 8. The power coefficients for translation (CP,t) and rotation (CP,r) versus time during one cycle (midstroke angle of attack m=35°, symmetrical rotation, non-dimensional duration of wing rotation r=0.36c). c, non-dimensional period of one flapping cycle; , non-dimensional time.

View larger version (25K): [in this window] [in a new window]   Fig. 9. Non-dimensional angular velocity of pitching rotation and azimuthal rotation (A), lift coefficient CL (B) and drag coefficient CD (C) versus time during one cycle for three different timings of wing rotation (midstroke angle of attack m=35°, non-dimensional duration of wing rotation r=0.36c). c, non-dimensional period of one flapping cycle; , non-dimensional time.

View larger version (25K): [in this window] [in a new window]   Fig. 12. The power coefficients for translation (CP,t) and rotation (CP,r) versus time during one cycle for the different wing rotation timings (midstroke angle of attack m=35°, non-dimensional duration of wing rotation r=0.36c). c, non-dimensional period of one flapping cycle; , non-dimensional time.

Dickinson et al. (1999) and Sun and Tang (2002) used _r0.36_c. In the present study, we first assumed _r0.36_c and then investigated the effects of varying _r. At this point, all the kinematic parameters had been determined except _m, which was determined using the force balance condition.

The calculated lift coefficients versus time for three values of _m are shown in Fig. 5. The mean lift coefficient plotted against _m is shown in Fig. 6. In the range of _m considered (_m=25-50°), increases with _m; at _m=35°, , which is the value needed to balance the weight of the insect. At _m=35°, the mean drag coefficient during an up- or downstroke (represented by ) was calculated to be 1.55, which is significantly larger than (=1.15). The lift-to-drag ratio is 1.15/1.55=0.74.

View larger version (24K): [in this window] [in a new window]   Fig. 5. Non-dimensional angular velocity of pitching rotation and azimuthal rotation (A), lift coefficient CL (B) and drag coefficient CD (C) versus time during one cycle for three midstroke angles of attack m (symmetrical rotation, non-dimensional duration of wing rotation r=0.36c). c, non-dimensional period of one flapping cycle; , non-dimensional time.

Power requirements
As shown above, at _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.

Aerodynamic torque
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.

Inertial torque
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 (Fig. 7C). Near the end of the stroke, the sign of C_Q,i,r is also opposite to that of . In this part of the stroke, although has the same sign as has the opposite sign to . In equation 37, is multiplied by I_xy, which is much larger than I_yy; thus, its effect dominates over that of other terms in the equation, leading to the above result. This shows that, for the flapping motion considered, the inertial torque of rotation will always contribute to `negative' work.

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 (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,t appear 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.

Specific power
The body-mass-specific power, denoted by P^*, is defined as the mean mechanical power over a flapping cycle (or a stroke in the case of normal hovering) divided by the mass of the insect. P^* can be written as follows:

(44)

where m is the mass of the insect and C_W is the coefficient of work per cycle (_L,w=1.15, _c=8.421 and U=2.19 ms^-1, as discussed above).

When calculating C_W, one needs to consider how the `negative' work fits into the power budget (Ellington, 1984c). There are three possibilities (Ellington, 1984c; Weis-Fogh, 1972, 1973). One is that the negative power is simply dissipated as heat and sound by some form of an end stop; it can then be ignored in the power budget. The second is that, during the period of negative work, the excess energy can be stored by an elastic element, and this energy can then be released when the wing does positive work. The third is that the flight muscles do negative work (i.e. they are stretched while developing tension, instead of contracting as in `positive' work), but the negative work uses much less metabolic energy than an equivalent amount of positive work. Of these three possibilities, we calculated C_W (or P^*) on the basis of the assumption that the muscles act as an end stop. C_W is written as:

(45)

It should be pointed out that, for the insect considered in the present study, the negative work is much smaller than the positive work (see Fig. 8 and below); therefore, C_W calculated by considering the other possibilities will not be very different from that calculated from equation 45.

Using C_W,t⁺ and C_W,r⁺ calculated above, C_W was 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 _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_D than 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 for the advanced rotation case is 1.47, 28 % larger than that for the symmetrical rotation case (1.15); for the delayed rotation case is only 0.39, which is 66 % smaller than that for the symmetrical rotation case.

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,r and C_Q,i,r are much smaller than C_Q,a,t and C_Q,i,t, respectively.

View larger version (18K): [in this window] [in a new window]   Fig. 10. Non-dimensional angular velocity of pitching rotation and azimuthal rotation (A), aerodynamic (B) and inertial (C) torque coefficients for translation (CQ,a,t and CQ,i,t, respectively) and rotation (CQ,a,r and CQ,i,r, respectively) versus time during one cycle (midstroke angle of attack m=35°, advanced rotation, non-dimensional duration of wing rotation r=0.36c). c, non-dimensional period of one flapping cycle; , non-dimensional time.

View larger version (18K): [in this window] [in a new window]   Fig. 11. Non-dimensional angular velocity of pitching rotation and azimuthal rotation (A), aerodynamic (B) and inertial (C) torque coefficients for translation (CQ,a,t and CQ,i,t, respectively) and rotation (CQ,a,r and CQ,i,r, respectively) versus time during one cycle (midstroke angle of attack m=35°, delayed rotation, non-dimensional duration of wing rotation r=0.36c). c, non-dimensional period of one flapping cycle; , non-dimensional time.

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,r and 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, is 28 % larger than that of the symmetrical rotation case. This shows that advanced rotation can produce more lift but is very energy-demanding. For the delayed rotation case, C_W is approximately 10 % larger than for the symmetrical rotation case but, as noted above, its is 66 % smaller; therefore, the energy spent per unit is much larger than for the symmetrical rotation case. The above results show that advanced rotation and delayed rotation would be much more costly if used in balanced, long-duration flight.

View this table: [in this window] [in a new window]   Table 1. Effects of the timing of wing rotation on lift and power (r=0.36c)

For reference, we calculated another case in which advanced rotation timing was employed in balanced flight but _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.

Effects of flip duration
In the above analyses, the duration of wing rotation (or flip duration) was taken as _r=0.36_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 durations _r=0.24_c and _r=0.19_c, with the above results for _r=0.36_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, _m was therefore adjusted. For _r=0.24_c, _m was changed to 36.5° to give a of 1.15; for _r=0.19_c, _m was changed to 38.5°.

View larger version (27K): [in this window] [in a new window]   Fig. 13. Non-dimensional angular velocity of pitching rotation and azimuthal rotation (A), lift coefficient CL (B) and drag coefficient CD (C) versus time during one cycle for three different values of the non-dimensional duration of wing rotation r. Symmetrical rotation; mean lift coefficient (midstroke angle of attack m was adjusted to make the mean lift equal to insect weight). c, non-dimensional period of one flapping cycle; , non-dimensional time.

From Fig. 13C, it can be seen that, when _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 smaller _r.

View larger version (25K): [in this window] [in a new window]   Fig. 14. The power coefficients for translation (CP,t) and rotation (CP,r) versus time during one cycle for three different values of the non-dimensional duration of wing rotation r. Symmetrical rotation; midstroke angle of attack m was adjusted to make the mean lift equal to insect weight. c, non-dimensional period of one flapping cycle; , non-dimensional time.

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. When _r is decreased to 0.24_c, C_W is 12.96, much smaller than for _r=0.36_c. When _r is further decreased to 0.19_c, C_W was slightly greater (13.06). This is because when _r is decreased to 0.19_c, C_W,t⁺ also decreases; however, C_W,r⁺ increases (possibly due to the wing rotation becoming very fast).

View this table: [in this window] [in a new window]   Table 2. Effects of duration of wing rotation on lift and power (m adjusted for balanced hovering flight)

For _r=0.24_c (which has approximately the same C_W as _r=0.19_c), the mass-specific power P^* was computed to be 28.7 W kg^-1. If the ratio of the flight muscle mass to the body mass is known, the power per unit flight muscle or muscle-mass-specific power (P_m^*) can be calculated from the body-mass-specific power. Lehmann and Dickinson (1997) obtained a value of 0.3 for the ratio for fruit fly Drosophila melanogaster. This value gives:

(46)

The muscle efficiency, , is:

(47)

where R_m is the body-mass-specific metabolic rate. Lehmann et al. (2000) measured the body-mass-specific CO₂ released for several species of fruit fly in the genus Drosophila. For D. virilis in hovering flight, the rate of CO₂ release was 30.1 ml g^-1 h^-1, corresponding to a body-mass-specific metabolic rate of 170 W kg^-1. The calculated muscle-mass-specific power and muscle efficiency are P_m^*=95.7 W kg^-1 and =17 %.

It is very interesting to look at the drag on the wing. In the above case (_r=0.24_c, _m=36.5°), the mean drag coefficient over an up- or downstroke is 1.46; . We see that, for this tiny hovering insect, its wings must overcome a drag that is 27 % larger than its weight to produce a lift that equals its weight. (This is very different from a large fast-flying bird, which only needs to overcome a drag that is a small fraction of its weight, and from a hovering helicopter, the blades of which need to overcome a drag that represents an even smaller fraction of its weight.)

Comparison between the calculated results and previous data
The above results showed that, for a duration of wing rotation _r=0.19_c and _r=0.24_c, the power expenditure for hovering flight in Drosoph