SUMMARY

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.

Introduction

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.

Fig. 1.

Sketches of the reference frames and wing motion. (A) OXYZ is an inertial frame, with the XY plane in the stroke plane. oxyzis a frame fixed on the wing, with the x axis along the wing chord,and the y axis along the wing span. ϕ is the azimuthal angle of the wing, α is the angle of attack and R is the wing length.(B) The motion of a section of the wing.

Fig. 1.

Sketches of the reference frames and wing motion. (A) OXYZ is an inertial frame, with the XY plane in the stroke plane. oxyzis a frame fixed on the wing, with the x axis along the wing chord,and the y axis along the wing span. ϕ is the azimuthal angle of the wing, α is the angle of attack and R is the wing length.(B) The motion of a section of the wing.

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.

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).

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 ut, which takes a constant value of Umexcept at the beginning and near the end of a stroke. During the acceleration at the beginning of a stroke, ut is given by:
\[\ u_{\mathrm{t}}^{+}=U_{\mathrm{m}}^{+}\mathrm{sin}[\frac{{\pi}({\tau}-{\tau}_{0})}{{\Delta}{\tau}_{\mathrm{t}}}];{\tau}_{0}{\leq}{\tau}{\leq}[{\tau}_{0}+(\frac{{\Delta}{\tau}_{\mathrm{t}}}{2})],\]
1
where ut+=Ut/U(U is the mean value of ut over a stroke and is used as reference velocity and ut+ is the non-dimensional translational velocity of the wing), Um+=Um/U(Um+ is the maximum of ut+), τ=tU/c (t is dimensional time and τ is the non-dimensional time), τ0 is the non-dimensional time at which the stroke starts andτ 0+(Δτt/2) is the time at which the acceleration at the beginning of the stroke finishes. Δτtis the duration of deceleration/acceleration around stroke reversal. Near the end of the stroke, the wing decelerates from ut+=Um+ to ut+=0 according to:
\[\ u_{\mathrm{t}}^{+}=U_{\mathrm{m}}^{+}\mathrm{sin}\left\{\frac{{\pi}}{{\Delta}{\tau}_{\mathrm{t}}}[{\tau}-{\tau}_{1}+(\frac{{\Delta}{\tau}_{\mathrm{t}}}{2})]\right\};{\tau}_{1}{\leq}{\tau}{\leq}[{\tau}_{1}+(\frac{{\Delta}{\tau}_{\mathrm{t}}}{2})],\]
2
where τ1 is the non-dimensional time at which the deceleration starts. The azimuth-rotational speed of the wing is related to ut. Denoting the azimuthal-rotational speed as
\({\dot{{\phi}}}\)
, we have
\({\dot{{\phi}}}({\tau})=\frac{{\mu}^{\mathrm{t}}}{r_{2}}\)
.
The angle of attack of the wing is denoted by α. It also takes a constant value except at the beginning or near the end of a stroke. The constant value is denoted by αm. Around the stroke reversal,α changes with time and the angular velocity
\({\dot{{\alpha}}}\)
is given by:
\[\ {\dot{{\alpha}}}^{+}=0.5{\dot{{\alpha}}}_{0}^{+}\{1-\mathrm{cos}[\frac{2{\pi}({\tau}-{\tau}_{\mathrm{r}})}{{\Delta}{\tau}_{\mathrm{r}}}]\};{\tau}_{\mathrm{r}}{\leq}{\tau}{\leq}({\tau}_{\mathrm{r}}+{\Delta}{\tau}_{\mathrm{r}}),\]
3
where the non-dimensional form
\({\dot{{\alpha}}}^{+}=\frac{{\dot{{\alpha}}}c}{U},{\ }{\dot{{\alpha}}}_{0}^{+}\)
is a constant, τr is the time at which the rotation starts andΔτ r is the time interval over which the rotation lasts. In the time interval Δτr, the wing rotates fromα=αm to α=180 °-αm. Therefore, when αm and Δτr are specified,
\({\dot{{\alpha}}}_{0}^{+}\)
can be determined (around the next stroke reversal, the wing would rotate fromα=180 °-αm to α=αm; 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.Δτ r is termed wing rotation duration (or flip duration),and τr is termed the rotation (or flip) timing, Whenτ r is chosen such that the majority of the wing rotation is conducted near the end of a stroke, it is called advanced rotation; whenτ 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 τr 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 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

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:
\[\ \frac{{\partial}u}{{\partial}X}+\frac{{\partial}v}{{\partial}Y}+\frac{{\partial}w}{{\partial}Z}=0,\]
4
\[\ \frac{{\partial}u}{{\partial}{\tau}}+u\frac{{\partial}u}{{\partial}X}+v\frac{{\partial}u}{{\partial}Y}+w\frac{{\partial}u}{{\partial}Z}=-\frac{{\partial}p}{{\partial}X}+\frac{1}{Re}\left(\frac{{\partial}^{2}u}{{\partial}X^{2}}+\frac{{\partial}^{2}u}{{\partial}Y^{2}}+\frac{{\partial}^{2}u}{{\partial}Z^{2}}\right),\]
5
\[\ \frac{{\partial}v}{{\partial}{\tau}}+u\frac{{\partial}v}{{\partial}X}+v\frac{{\partial}v}{{\partial}Y}+w\frac{{\partial}v}{{\partial}Z}=-\frac{{\partial}p}{{\partial}Y}+\frac{1}{Re}\left(\frac{{\partial}^{2}v}{{\partial}X^{2}}+\frac{{\partial}^{2}v}{{\partial}Y^{2}}+\frac{{\partial}^{2}v}{{\partial}Z^{2}}\right),\]
6
\[\ \frac{{\partial}w}{{\partial}{\tau}}+u\frac{{\partial}w}{{\partial}X}+v\frac{{\partial}w}{{\partial}Y}+w\frac{{\partial}w}{{\partial}Z}=-\frac{{\partial}p}{{\partial}Z}+\frac{1}{Re}\left(\frac{{\partial}^{2}w}{{\partial}X^{2}}+\frac{{\partial}^{2}w}{{\partial}Y^{2}}+\frac{{\partial}^{2}w}{{\partial}Z^{2}}\right),\]
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 Yaxis 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 α and ϕ (defined in Fig. 1A), the relationship between these two coordinate systems is given by:
\[\ \left(\begin{array}{c}x\\y\\z\end{array}\right)\begin{array}{c}=\\=\\=\end{array}\left[\begin{array}{ccc}\mathrm{cos}{\alpha}\mathrm{cos}{\phi}&\mathrm{sin}{\phi}\mathrm{cos}{\alpha}&-\mathrm{sin}{\alpha}\\-\mathrm{sin}{\phi}&\mathrm{cos}{\phi}&0\\\mathrm{sin}{\alpha}\mathrm{cos}{\phi}&\mathrm{sin}{\alpha}\mathrm{sin}{\phi}&\mathrm{cos}{\alpha}\end{array}\right]\left(\begin{array}{c}X\\Y\\Z\end{array}\right),\]
8
and
\[\ \left(\begin{array}{c}X\\Y\\Z\end{array}\right)\begin{array}{c}=\\=\\=\end{array}\left[\begin{array}{ccc}\mathrm{cos}{\alpha}\mathrm{cos}{\phi}&-\mathrm{sin}{\phi}&\mathrm{cos}{\phi}\mathrm{sin}{\alpha}\\\mathrm{sin}{\phi}\mathrm{cos}{\alpha}&\mathrm{cos}{\alpha}&\mathrm{sin}{\phi}\mathrm{sin}{\alpha}\\-\mathrm{sin}{\alpha}&0&\mathrm{cos}{\alpha}\end{array}\right]\left(\begin{array}{c}x\\y\\z\end{array}\right).\]
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 CLand CD, respectively, are defined as follows:
\[\ C_{\mathrm{L}}=\frac{L}{0.5{\rho}U^{2}S},\]
10
\[\ C_{\mathrm{D}}=\frac{D}{0.5{\rho}U^{2}S},\]
11
where ρ 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 -Ma, can be written as follows (assuming that the thickness of the wing is very small):
\[\ \begin{array}{rl}-\mathbf{M}_{\mathbf{\mathrm{a}}}&={{\int}_{_{S}}}\mathbf{r}{\times}\mathbf{F}\mathrm{dS}\\&{\approx}-{{\int}_{_{S}}}(yf_{\mathrm{z}}\mathbf{i}-xf_{\mathrm{z}}\mathbf{j}-yf_{\mathrm{x}}\mathbf{k})\mathrm{d}S,\end{array}\]
12
where S is the wing surface area, F is the aerodynamic force in unit wing area, r is the distance vector, fx, fy and fz are the three components of F in the oxyz coordinate system, x, y and zare 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, fy, the component of F along the wing span, is neglected and fx and fz can be obtained from the solution of the flow equations. The angular velocity vector of the wing, Ω, has the following three components in the oxyzcoordinate system:
\[\ \mathbf{{\Omega}}=(-{\dot{{\phi}}}\mathrm{sin}{\alpha},{\dot{{\alpha}}},{\dot{{\phi}}}\mathrm{cos}{\alpha}).\]
13
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.

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 Pa, can be written as:
\[\ \begin{array}{rl}P_{\mathrm{a}}&=\mathbf{M}_{\mathbf{\mathrm{a}}}{\cdot}\mathbf{{\Omega}}\\&={{\int}_{_{S}}}(yf_{\mathrm{z}}{\dot{{\phi}}}\mathrm{sin}{\alpha}+xf_{\mathrm{z}}{\dot{{\alpha}}}+yf_{\mathrm{x}}{\dot{{\phi}}}\mathrm{cos}{\alpha})\mathrm{d}S\\&=Q_{\mathrm{a,t}}{\dot{{\phi}}}+Q_{\mathrm{a,r}}{\dot{{\alpha}}},\end{array}\]
14
where
\[\ Q_{\mathrm{a,t}}={{\int}_{_{S}}}y(f_{\mathrm{z}}\mathrm{sin}{\alpha}+yf_{\mathrm{x}}\mathrm{cos}{\alpha})\mathrm{d}S,\]
15
\[\ Q_{\mathrm{a,r}}={{\int}_{_{S}}}xf_{\mathrm{z}}\mathrm{d}S.\]
16
Qa,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. Qar 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:
\[\ C_{\mathrm{Q,a,t}}=\frac{Q_{\mathrm{a,t}}}{0.5{\rho}U^{2}Sc};\]
17
\[\ C_{\mathrm{Q,a,r}}=\frac{Q_{\mathrm{a,r}}}{0.5{\rho}U^{2}Sc}.\]
18
The aerodynamic power coefficient is defined as:
\[\ C_{\mathrm{P,a}}=\frac{P_{\mathrm{a}}}{0.5{\rho}U^{3}S}\]
19
and can be written as:
\[\ C_{\mathrm{P,a}}=C_{\mathrm{Q,a,t}}{\dot{{\phi}}}^{+}+C_{\mathrm{Q,a,r}}{\dot{{\alpha}}}^{+},\]
20
where
\({\dot{{\phi}}}^{+}\)
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:
\[\ I_{\mathrm{xx}}={{\int}}(y^{2}+z^{2})\mathrm{d}m_{\mathrm{w}}{\approx}{{\int}}y^{2}\mathrm{d}m_{\mathrm{w}},\]
21
\[\ I_{\mathrm{yy}}={{\int}}(z^{2}+x^{2})\mathrm{d}m_{\mathrm{w}}{\approx}{{\int}}x^{2}\mathrm{d}m_{\mathrm{w}},\]
22
\[\ I_{\mathrm{zz}}={{\int}}(x^{2}+y^{2})\mathrm{d}m_{\mathrm{w}},\]
23
\[\ I_{\mathrm{xy}}={{\int}}xy\mathrm{d}m_{\mathrm{w}},\]
24
\[\ I_{\mathrm{xz}}={{\int}}xz\mathrm{d}m_{\mathrm{w}}{\approx}0,\]
25
\[\ I_{\mathrm{yz}}={{\int}}yz\mathrm{d}m_{\mathrm{w}}{\approx}0,\]
26
where dmw is a mass element of the wing. The angular momentum of the wing H and its three components in the oxyzcoordinate system are as follows:
\[\ \mathbf{H}=-\mathbf{i}(I_{\mathrm{xx}}{\dot{{\phi}}}\mathrm{sin}{\alpha}+I_{\mathrm{xy}}{\dot{{\alpha}}})+\mathbf{j}(I_{\mathrm{yy}}{\dot{{\alpha}}}+I_{\mathrm{xy}}{\dot{{\phi}}}\mathrm{sin}{\alpha})+\mathbf{k}I_{\mathrm{zz}}{\dot{{\phi}}}\mathrm{cos}{\alpha},\]
27
where i, j and k are unit vectors in the x, y and z directions, respectively. The inertial moment of the wing, Mi, is:
\[\ \begin{array}{rl}\mathbf{M}_{\mathbf{\mathrm{i}}}&=\left(\frac{\mathrm{d}\mathbf{H}}{\mathrm{d}t}\right)_{\mathrm{xyz}}+\mathbf{{\Omega}}{\times}\mathbf{H}\\&=m_{\mathrm{i}}^{\mathrm{x}}\mathbf{i}+m_{\mathrm{i}}^{\mathrm{y}}\mathbf{j}+m_{\mathrm{i}}^{\mathrm{z}}\mathbf{k},\end{array}\]
28
where (dH/dt)xyz denotes the time derivative of the vector as observed in the rotating system and mix, miy and miz, the x, y and zcomponents of Mi, respectively, are as follows:
\[\ m_{\mathrm{i}}^{\mathrm{x}}=-\mathbf{i}[I_{\mathrm{xx}}{\ddot{{\phi}}}\mathrm{sin}{\alpha}+I_{\mathrm{xy}}({\dot{{\phi}}}^{2}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha}+{\ddot{{\alpha}}})],\]
29
\[\ m_{\mathrm{i}}^{\mathrm{y}}=\mathbf{j}(I_{\mathrm{yy}}{\ddot{{\alpha}}}+I_{\mathrm{xy}}{\ddot{{\phi}}}\mathrm{sin}{\alpha}+I_{\mathrm{yy}}{\dot{{\phi}}}^{2}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha}),\]
30
\[\ m_{\mathrm{i}}^{\mathrm{z}}=\mathbf{k}[I_{\mathrm{zz}}{\ddot{{\phi}}}\mathrm{cos}{\alpha}-2I_{\mathrm{yy}}{\dot{{\phi}}}{\dot{{\alpha}}}\mathrm{sin}{\alpha}+I_{\mathrm{xy}}({\dot{{\alpha}}}^{2}-{\dot{{\phi}}}\mathrm{sin}^{2}{\alpha})],\]
31
where
\({\ddot{{\phi}}}\)
and
\({\ddot{{\alpha}}}\)
are the angular acceleration of the azimuthal and pitching rotations, respectively. In deriving equation 28, IyyIzz-Izz was used. The inertial power of the wing is written as follows:
\[\ \begin{array}{rl}P_{\mathrm{i}}&=\mathbf{M}_{\mathbf{i}}{\cdot}\mathbf{{\Omega}}\\&=-m_{\mathrm{i}}^{\mathrm{x}}{\dot{{\phi}}}\mathrm{sin}{\alpha}+m_{\mathrm{i}}^{\mathrm{y}}{\dot{{\alpha}}}+m_{\mathrm{i}}^{\mathrm{z}}{\dot{{\phi}}}\mathrm{cos}{\alpha},\\&=(-m_{\mathrm{i}}^{\mathrm{x}}\mathrm{sin}{\alpha}+m_{\mathrm{i}}^{\mathrm{z}}\mathrm{cos}{\alpha}){\dot{{\phi}}}+m_{\mathrm{i}}^{\mathrm{y}}{\dot{{\alpha}}}\\&=Q_{\mathrm{i,t}}{\dot{{\phi}}}+Q_{\mathrm{i,r}}{\dot{{\alpha}}},\end{array}\]
32
where
\[\ \begin{array}{rl}Q_{\mathrm{i,t}}&=-m_{\mathrm{i}}^{\mathrm{x}}\mathrm{sin}{\alpha}+m_{\mathrm{i}}^{\mathrm{z}}\mathrm{cos}{\alpha}\\&=(I_{\mathrm{xx}}\mathrm{sin}^{2}{\alpha}+I_{\mathrm{zz}}\mathrm{cos}^{2}{\alpha}){\ddot{{\phi}}}+I_{\mathrm{xy}}({\dot{{\alpha}}}^{2}\mathrm{cos}{\alpha}+{\ddot{{\alpha}}}\mathrm{sin}{\alpha})-2I_{\mathrm{yy}}{\dot{{\phi}}}{\dot{{\alpha}}}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha};\end{array}\]
33
\[\ \begin{array}{rl}Q_{\mathrm{i,r}}&=m_{\mathrm{i}}^{\mathrm{y}}\\&=I_{\mathrm{yy}}{\ddot{{\alpha}}}+I_{\mathrm{xy}}{\ddot{{\phi}}}\mathrm{sin}{\alpha}+I_{\mathrm{yy}}{\dot{{\phi}}}^{2}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha}.\end{array}\]
34
Qi,t is the inertial torque around the axis of azimuthal rotation and is termed the inertial torque for translation. Qi,r is the inertial torque around the axis of pitching rotation and is termed the inertial torque for rotation. The coefficients of Pi, Qi,t and Qi,rare denoted by CP,i, CQ,i,t and CQ,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:
\[\ C_{\mathrm{P,i}}=C_{\mathrm{Q,i,t}}{\dot{{\phi}}}^{+}+C_{\mathrm{Q,i,r}}{\dot{{\alpha}}}^{+}\]
35
and the expressions for CQ,i,t and CQ,i,r are as follows:
\[\ C_{\mathrm{Q,i,t}}=\frac{I_{\mathrm{zz}}}{0.5{\rho}Sc^{3}}\left\{\left[\left(\mathrm{cos}^{2}{\alpha}+\frac{I_{\mathrm{xx}}}{I_{\mathrm{zz}}}\mathrm{sin}^{2}{\alpha}\right){\ddot{{\phi}}}^{+}\right]+\frac{I_{\mathrm{xy}}}{I_{\mathrm{zz}}}({\dot{{\alpha}}}^{+2}\mathrm{cos}{\alpha}+{\ddot{{\alpha}}}^{+}\mathrm{sin}{\alpha})-2\frac{I_{\mathrm{yy}}}{I_{\mathrm{zz}}}{\dot{{\phi}}}^{+}{\dot{{\alpha}}}^{+}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha}\right\};\]
36
\[\ C_{\mathrm{Q,i,r}}=\frac{I_{\mathrm{yy}}}{0.5{\rho}Sc^{3}}\left({\ddot{{\alpha}}}^{+}+\frac{I_{\mathrm{xy}}}{I_{\mathrm{yy}}}{\ddot{{\phi}}}^{+}\mathrm{sin}{\alpha}+{\dot{{\phi}}}^{+2}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha}\right),\]
37
where
\({\ddot{{\phi}}}^{+}\)
and
\({\ddot{{\alpha}}}^{+}\)
are the non-dimensional angular acceleration of the azimuthal and pitching rotations,respectively. Using the values of Ixx, Iyy, Izz and Ixygiven in the next section, CQ,i,t and CQ,i,r can be written as:
\[\ C_{\mathrm{Q,i,t}}=4.6[(\mathrm{cos}^{2}{\alpha}+0.91\mathrm{sin}^{2}{\alpha}){\ddot{{\phi}}}^{+}+0.19({\dot{{\alpha}}}^{+2}\mathrm{cos}{\alpha}+{\ddot{{\alpha}}}^{+}\mathrm{sin}{\alpha})-0.17{\dot{{\phi}}}^{+}{\dot{{\alpha}}}^{+}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha}],\]
38
\[\ C_{\mathrm{Q,i,r}}=0.40({\ddot{{\alpha}}}^{+}+2.14{\ddot{{\phi}}}^{+}\mathrm{sin}{\alpha}+{\dot{{\phi}}}^{+2}\mathrm{sin}{\alpha}\mathrm{cos}{\alpha}).\]
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 Ixy is over twice as large as Iyy, the contribution from the translational acceleration
\({\ddot{{\phi}}}^{+}\)
to the rotational inertial torque can be larger than that from the rotational acceleration
\({\ddot{{\alpha}}}^{+}\)
.

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:
\[\ P=(\mathbf{M}_{\mathbf{\mathrm{a}}}+\mathbf{M}_{\mathbf{\mathrm{i}}}){\cdot}\mathbf{{\Omega}}.\]
40
Henceforth, it is termed simply the power. Combining equations 20 and 35, the non-dimensional power coefficient (represented by CP), can be written as follows:
\[\ C_{\mathrm{P}}=C_{\mathrm{P,t}}+C_{\mathrm{P,r}}\]
41
where
\[\ C_{\mathrm{P,t}}=(C_{\mathrm{Q,a,t}}+C_{\mathrm{Q,i,t}}){\dot{{\phi}}}^{+},\]
42
\[\ C_{\mathrm{P,r}}=(C_{\mathrm{Q,a,r}}+C_{\mathrm{Q,i,r}}){\dot{{\alpha}}}^{+}.\]
43
CP,t is the coefficient of power for translation and CP,r the coefficient of power for rotation.

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 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; L,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 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: 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.

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. τ, non-dimensional time;τ c, non-dimensional period of one flapping cycle.

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. τ, non-dimensional time;τ c, non-dimensional period of one flapping cycle.

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.

Fig. 6.

Mean lift coefficient

\(C_{\mathrm{L}}\)
versusmidstroke angle of attack αm (symmetrical rotation,non-dimensional duration of wing rotationΔτ r=0.36τc, where τc is the non-dimensional period of one flapping cycle).

Fig. 6.

Mean lift coefficient

\(C_{\mathrm{L}}\)
versusmidstroke angle of attack αm (symmetrical rotation,non-dimensional duration of wing rotationΔτ r=0.36τc, where τc is the non-dimensional period of one flapping cycle).

Fig. 7.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, symmetrical rotation, non-dimensional duration of wing rotation Δτr=0.36τc).τ c, non-dimensional period of one flapping cycle; τ,non-dimensional time.

Fig. 7.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, symmetrical rotation, non-dimensional duration of wing rotation Δτr=0.36τc).τ c, non-dimensional period of one flapping cycle; τ,non-dimensional time.

Fig. 8.

The power coefficients for translation (CP,t) and rotation (CP,r) versus time during one cycle(midstroke angle of attack αm=35°, symmetrical rotation,non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

Fig. 8.

The power coefficients for translation (CP,t) and rotation (CP,r) versus time during one cycle(midstroke angle of attack αm=35°, symmetrical rotation,non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

Fig. 9.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

Fig. 9.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

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 attackα m=35°, non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

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 attackα m=35°, non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

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

\(C_{\mathrm{L}}\)
plotted against αm is shown in Fig. 6. In the range ofα m considered (αm=25-50°),
\(C_{\mathrm{L}}\)
increases with αm; atα m=35°,
\(C_{\mathrm{L}}=1.15\)
, which is the value needed to balance the weight of the insect. Atα m=35°, the mean drag coefficient during an up- or downstroke (represented by
\(C_{\mathrm{D}}\)
) was calculated to be 1.55, which is significantly larger than
\(C_{\mathrm{L}}\)
(=1.15). The lift-to-drag ratio is 1.15/1.55=0.74.

Fig. 5.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(A), lift coefficient CL (B) and drag coefficient CD (C) versus time during one cycle for three midstroke angles of attack αm (symmetrical rotation,non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

Fig. 5.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(A), lift coefficient CL (B) and drag coefficient CD (C) versus time during one cycle for three midstroke angles of attack αm (symmetrical rotation,non-dimensional duration of wing rotationΔτ r=0.36τc). τc,non-dimensional period of one flapping cycle; τ, non-dimensional time.

Power requirements

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.

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, 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.

Inertial torque

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

\({\dot{{\alpha}}}^{+}\)
(Fig. 7C). Near the end of the stroke, the sign of CQ,i,r is also opposite to that of
\({\dot{{\alpha}}}^{+}\)
. In this part of the stroke, although
\({\ddot{{\alpha}}}^{+}\)
has the same sign as
\({\dot{{\alpha}}}^{+},{\ddot{{\phi}}}^{+}\)
has the opposite sign to
\({\dot{{\alpha}}}^{+}\)
. In equation 37,
\({\ddot{{\phi}}}^{+}\)
is multiplied by Ixy, which is much larger than Iyy;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 (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.

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:
\[\ \begin{array}{c}P^{*}=\frac{0.5{\rho}U^{3}S_{\mathrm{t}}(\frac{C_{\mathrm{W}}}{{\tau}_{\mathrm{c}}})}{m}\\=\frac{9.81UC_{\mathrm{W}}}{({\tau}_{\mathrm{c}}{\bar{C}}_{\mathrm{L,w}})},\end{array}\]
44
where m is the mass of the insect and CW is the coefficient of work per cycle(L,w=1.15,τ c=8.421 and U=2.19 ms-1, as discussed above).
When calculating CW, 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 CW (or P*) on the basis of the assumption that the muscles act as an end stop. CW is written as:
\[\ C_{\mathrm{W}}=C_{\mathrm{W,t}}^{+}+C_{\mathrm{W,r}}^{+}.\]
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, CW calculated by considering the other possibilities will not be very different from that calculated from equation 45.

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

\(C_{\mathrm{L}}\)
for the advanced rotation case is 1.47, 28 % larger than that for the symmetrical rotation case (1.15);
\(C_{\mathrm{L}}\)
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 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.

Fig. 10.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, advanced rotation, non-dimensional duration of wing rotation Δτr=0.36τc).τ c, non-dimensional period of one flapping cycle; τ,non-dimensional time.

Fig. 10.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, advanced rotation, non-dimensional duration of wing rotation Δτr=0.36τc).τ c, non-dimensional period of one flapping cycle; τ,non-dimensional time.

Fig. 11.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, delayed rotation, non-dimensional duration of wing rotation Δτr=0.36τc).τ c, non-dimensional period of one flapping cycle; τ,non-dimensional time.

Fig. 11.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 attackα m=35°, delayed rotation, non-dimensional duration of wing rotation Δτr=0.36τc).τ c, non-dimensional period of one flapping cycle; τ,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). 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,

\(C_{\mathrm{L}}\)
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, CW is approximately 10 % larger than for the symmetrical rotation case but, as noted above, its
\(C_{\mathrm{L}}\)
is 66 % smaller; therefore, the energy spent per unit
\(C_{\mathrm{L}}\)
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.

Table 1.

Effects of the timing of wing rotation on lift and power(Δτr=0.36τc)

αm (degrees)Rotation timing
\(C_{\mathrm{L}}\)
CW,t+CW,t-CW,r+CW,r-CW
35 Symmetrical 1.15 15.96 -1.0 0.56 -2.30 16.52 
35 Advanced 1.47 26.08 -1.62 3.48 -0.72 29.56 
35 Delayed 0.39 17.70 -1.90 0.14 -2.70 17.84 
22 Advanced 1.15 24.74 -1.84 5.36 -0.40 30.10 
αm (degrees)Rotation timing
\(C_{\mathrm{L}}\)
CW,t+CW,t-CW,r+CW,r-CW
35 Symmetrical 1.15 15.96 -1.0 0.56 -2.30 16.52 
35 Advanced 1.47 26.08 -1.62 3.48 -0.72 29.56 
35 Delayed 0.39 17.70 -1.90 0.14 -2.70 17.84 
22 Advanced 1.15 24.74 -1.84 5.36 -0.40 30.10 

\(C_{\mathrm{L}}\)
⁠, mean lift coefficient; CW, coefficient of work per cycle; CW,t+ and CW,t-,coefficients of positive and negative work for translation, respectively; CW,r+ and CW,r-,coefficients of positive and negative work for rotation, respectively;α m, midstroke angle of attack; Δτr,non-dimensional duration of wing rotation; τc, non-dimensional period of one flapping cycle.

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

\(C_{\mathrm{L}}\)
of 1.15; forΔτ r=0.19τc, αm was changed to 38.5°.

Fig. 13.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 rotationΔτ r. Symmetrical rotation; mean lift coefficient
\(C_{\mathrm{L}}=1.15\)
(midstroke angle of attackα m was adjusted to make the mean lift equal to insect weight). τc, non-dimensional period of one flapping cycle;τ, non-dimensional time.

Fig. 13.

Non-dimensional angular velocity of pitching rotation

\({\dot{{\alpha}}}^{+}\)
and azimuthal rotation
\({\dot{{\phi}}}^{+}\)
(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 rotationΔτ r. Symmetrical rotation; mean lift coefficient
\(C_{\mathrm{L}}=1.15\)
(midstroke angle of attackα m was adjusted to make the mean lift equal to insect weight). τc, non-dimensional period of one flapping cycle;τ, non-dimensional time.

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.

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 rotationΔτ r. Symmetrical rotation; midstroke angle of attackα m was adjusted to make the mean lift equal to insect weight.τ c, non-dimensional period of one flapping cycle; τ,non-dimensional time.

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 rotationΔτ r. Symmetrical rotation; midstroke angle of attackα m was adjusted to make the mean lift equal to insect weight.τ c, non-dimensional period of one flapping cycle; τ,non-dimensional time.

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).

Table 2.

Effects of duration of wing rotation on lift and power(αm adjusted for balanced hovering flight)

Δτrcαm (degrees)
\(C_{\mathrm{L}}\)
CW,t+CW,t-CW,r+CW,r-CW
0.36 35 1.15 15.96 -1.00 0.56 -2.30 16.52 
0.24 36.5 1.15 12.70 -0.66 0.26 -1.76 12.96 
0.19 36.5 1.15 12.08 -0.88 0.98 -1.42 13.06 
Δτrcαm (degrees)
\(C_{\mathrm{L}}\)
CW,t+CW,t-CW,r+CW,r-CW
0.36 35 1.15 15.96 -1.00 0.56 -2.30 16.52 
0.24 36.5 1.15 12.70 -0.66 0.26 -1.76 12.96 
0.19 36.5 1.15 12.08 -0.88 0.98 -1.42 13.06 

\(C_{\mathrm{L}}\)
⁠, mean lift coefficient; CW, coefficient of work per cycle; CW,t+ and CW,t-,coefficients of positive and negative work for translation, respectively; CW,r+ and CW,r-,coefficients of positive and negative work for rotation, respectively;α m, midstroke angle of attack; Δτr,non-dimensional duration of wing rotation; τc, non-dimensional period of one flapping cycle.

For Δτr=0.24τc (which has approximately the same CW asΔτ r=0.19τc), 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 (Pm*) 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:
\[\ P_{\mathrm{m}}^{*}=\frac{P^{*}}{0.3}.\]
46
The muscle efficiency, η, is:
\[\ {\eta}=\frac{P^{*}}{R_{\mathrm{m}}},\]
47
where Rm 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 Pm*=95.7 W kg-1 and η=17 %.

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

\(C_{\mathrm{D}}\)
over an up- or downstroke is 1.46;
\(\frac{C_{\mathrm{L}}}{C_{\mathrm{D}}}=\frac{1.15}{\frac{1}{46}}=0.79\)
. 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 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

\({\dot{{\alpha}}}^{+}\)
is approximately 2,as seen in Fig. 13A). This mean rotational velocity is close to the value of 0.95 measured in the free forward flight of Drosophila melanogaster [data in Table 4 of Ennos(1989) multiplied by R/r2=1/0.58 because a different reference velocity was used]. It is also close to that measured in free hovering flight in craneflies, hoverflies and droneflies: approximately 0.9, 1.2 and 0.9,respectively [data in Table 2of Ellington (1984a) multiplied by 1/0.58]. Therefore, both from measurements from similar insects and from the calculated efficiency, it is reasonable to assume aΔτ r of approximately 20 % of τc.

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.

    List of symbols
     
  • c

    mean chord length

  •  
  • CD

    drag coefficient

  •  
  • \(C_{\mathrm{D}}\)

    mean drag coefficient (over an up- or downstroke)

  •  
  • CL

    lift coefficient

  •  
  • \(C_{\mathrm{L}}\)

    mean lift coefficient

  •  
  • L,w

    mean lift coefficient for supporting the insect's weight

  •  
  • CP

    power coefficient

  •  
  • CP,a

    coefficient of aerodynamic power

  •  
  • CP,i

    coefficient of inertial power

  •  
  • CP,r

    coefficient of power for rotation

  •  
  • CP,t

    coefficient of power for translation

  •  
  • CQ,a,r

    coefficient of aerodynamic torque for rotation

  •  
  • CQ,a,t

    coefficient of aerodynamic torque for translation

  •  
  • CQ,i,r

    coefficient of inertial torque for rotation

  •  
  • CQ,i,t

    coefficient of inertial torque for translation

  •  
  • CW

    coefficient of work per cycle

  •  
  • CW,r+

    coefficient of positive work for rotation

  •  
  • CW,r-

    coefficient of negative work for rotation

  •  
  • CW,t+

    coefficient of positive work for translation

  •  
  • CW,t-

    coefficient of negative work for translation

  •  
  • D

    drag

  •  
  • dmw

    mass element of the wing

  •  
  • F

    aerodynamic force per unit wing area

  •  
  • fx, fy, fz

    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

  •  
  • 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

  •  
  • L

    lift

  •  
  • m

    mass of the insect

  •  
  • mw

    wing mass of the insect

  •  
  • Ma

    aerodynamic moment

  •  
  • Mi

    inertial moment

  •  
  • mix, miy, miz

    x, y and z components of Mi,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

  •  
  • Pa

    aerodynamic power

  •  
  • Pi

    inertial power

  •  
  • P*

    body-mass-specific power

  •  
  • Pm*

    muscle-mass-specific power

  •  
  • Qa,r

    aerodynamic torque for rotation

  •  
  • Qa,t

    aerodynamic torque for translation

  •  
  • Qi,r

    inertial torque for rotation

  •  
  • Qi,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

  •  
  • Rm

    body-mass-specific metabolic rate

  •  
  • S

    area of one wing

  •  
  • St

    area of a wing pair

  •  
  • t

    time

  •  
  • u, v, w

    three components of non-dimensional fluid velocity

  •  
  • ut

    translational velocity of the wing

  •  
  • ut+

    non-dimensional translational velocity of the wing

  •  
  • U

    reference velocity (ut averaged over a stroke)

  •  
  • Um

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

  •  
  • Um+

    maximum of ut+

  •  
  • 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

Acknowledgements

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.

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