Effects of unsteady deformation of flapping wing on its aerodynamic forces

Transcription

1 Appl. Math. Mech. -Engl. Ed., 2008, 29(6): DOI /s c Editorial Committee of Appl. Math. Mech. and Springer-Verlag 2008 Applied Mathematics and Mechanics (English Edition) Effects of unsteady deformation of flapping wing on its aerodynamic forces DU Gang ( ) 1, SUN Mao ( ) 2 (1. Aeroengine Numerical Simulation Research Center, Beijing University of Aeronautics and Astronautics, Beijing , P. R. China; 2. Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing , P. R. China) (Communicated by ZHOU Zhe-wei) Abstract Effects of unsteady deformation of a flapping model insect wing on its aerodynamic force production are studied by solving the Navier-Stokes equations on a dynamically deforming grid. Aerodynamic forces on the flapping wing are not much affected by considerable twist, but affected by camber deformation. The effect of combined camber and twist deformation is similar to that of camber deformation. With a deformation of 6% camber and 20 twist (typical values observed for wings of many insects), lift is increased by 10% 20% and lift-to-drag ratio by around 10% compared with the case of a rigid flat-plate wing. As a result, the deformation can increase the maximum lift coefficient of an insect, and reduce its power requirement for flight. For example, for a hovering bumblebee with dynamically deforming wings (6% camber and 20 twist), aerodynamic power required is reduced by about 16% compared with the case of rigid wings. Key words insect, wing deformation, unsteady aerodynamic force, computational fluid dynamics Chinese Library Classification O Mathematics Subject Classification 76Z10 Introduction Recently, much progress has been made in our understanding of the unsteady aerodynamics of insect flapping wings. Dickinson and Götz [1] measured the aerodynamic forces of an airfoil started rapidly at high angles of attack in the Reynolds number (Re) range of fruit fly wing (Re = ), and showed that lift was enhanced by the presence of a leading edge vortex (LEV) or the delayed stall effect. After the initial start, lift coefficient as high as 2 was maintained within approximately 2 3 chord lengths of travel. Afterwards, lift drops due to the shedding of the LEV. Ellington, et al. [2] discovered that the LEVs exist on the wings of the hawkmoth Manduca Sexta and they do not shed in the translational phase of a half-stroke Received Jan. 17, 2007 / Revised Apr. 17, 2008 Project supported by the Fan Zhou Youth Science Fund of Beijing University of Aeronautics and Astronautics (No ) Corresponding author DU Gang, Doctor, dugang@buaa.edu.cn

2 732 DU Gang and SUN Mao (downstroke or upstroke), i.e., the delayed stall effect could be maintained in an entire halfstroke. This was confirmed by Willmott, et al. [3] using a mechanical model of the hawkmoth wings. Analysis of the momentum imparted to fluid by the vortex wake showed that the LEV could produce enough lift for insect-weight support. This high-lift mechanism was termed delayed stall mechanism. The delayed stall mechanism was further confirmed by computational fluid dynamics analysis, e.g., by Liu, et al. [4] and Sun and Tang [5]. As a result of the above works and many others (e.g., Refs. [5 14]), we are now better able to understand how insects produce large lift. In most of the above studies, rigid model wings were employed. However, observations showed that insect wings during flapping motion had noticeable deformations. Ellington [8], using a high-speed camera, made extensive studies on the wing kinematics of more than ten species of insects (flies and mosquitoes, bees, beetles and moths). He observed that the wings were gently cambered on both half-strokes (the downstroke and the upstroke) and that during the middle portion of each downstroke or upstroke, the angle of attack was greatest at the wing base and decreased towards the wing tip, i.e., the wing was twisted. Similar deformations were also observed by other authors, e.g., Ennos [11] and Dudley [12]. They estimated that the camber was several percent of the chord and the twist was around 10. Recently, Wang, et al. [13] conducted measurements on the deformation of dragonfly wings and found that camber was around 8% of the chord and that along the outer 80% of the wing length, the wing was twisted around 25. Ennos [14] explained why the insect wing developed twist and camber during flapping motion using an model insect wing. In his model wing, spars of corrugated membrane which incorporated stiffening veins branched serially from a V-section leading edge spar (Fig. 1(a)). The open, corrugated spars possessed great resistance to bending but were compliant in torsion. Torsion of the leading edge spar would result in torsion and relative movement of the rear spars. As a result, camber would automatically be set up in the wing as it twisted (Fig. 1(b)). Ennos and Wootton [15] observed that in some insects, the wing flexed along an oblique curve line and camber was created in the distal part of the wing (see Ref. [16] for explanation of why such a flexion causes camber). How do these deformations (camber and torsion) influence the aerodynamic forces and moments on insect wings in flapping motion? Some studies have been conducted on the effects of camber under steady-state conditions. Vogel [17] measured the aerodynamic forces on flat and cambered model fruit fly wings in a wind-tunnel. Dudley and Ellington [18] did a similar experiment on bumblebee wings. They showed that camber increased the lift coefficient and the lift/drag ratio. To the author s knowledge, for insect wings in flapping motion, the effects of twist and camber have not been studied. In the present work, the aerodynamic properties of a flapping model insect wing with time-varying twist and camber are studied, using the method of solving the Navier-Stokes equations on a dynamically deforming grid. Fig. 1 (a) Sketches of the wing of the damselfly Pyrrhosoma nymphula and the basic model of the wing (b) A sketch of the deformation of the model wing (drawn from Ref. [14])

3 Effects of unsteady deformation of flapping wing on its aerodynamic forces Computational method 1.1 Governing equations, geometric conservation law and numerical procedure The governing equations employed in this study are the unsteady three-dimensional incompressible Navier-Stokes equations, written in the inertial coordinate system oxyz and then transformed into the curvilinear coordinate system (ξ, η, ζ, τ) using a general time-dependent transformation in the form: ξ = ξ(x, y, z, t), η = η(x, y, z, t), ζ = ζ(x, y, z, t), τ = t. (1) The transformed equations written in conservative form are as follows: ( ) A + ( ) B + ( ) C = 0, (2) ξ J η J ζ J ˆq τ + ξ (ê ê ν) + ( ) ˆf ˆfν + η ζ (ĝ ĝ ν) = h GCL, (3) where J is the Jacobian of the transformation, h GCL is a term to enforce the geometric conservation law (GCL) for moving/deforming meshes [19], and A = ξ x u + ξ y v + ξ z w, B = η x u + η y v + η z w, C = ζ x u + ζ y v + ζ z w, ˆq = 1 (u, v, w)t J where ξ x = ξ x, etc. representative the metrics of the transformation; the expressions of ê, ˆf, etc. follow Refs. [21] and [22]. h GCL is defined as [ h GCL = q τ ( 1 J ) + (ξ τ J ) ξ + ( η τ J ) η + ( ζ ] τ J ) ζ. (4) A discussion and derivation of this term is given below. Upon transforming the Navier-Stokes equations in oxyz coordinate to the (ξ, η, ζ, τ) coordinates with the aid of the chain rule for partial derivatives, we have ˆq ( ) ˆf ˆfν + ζ (ĝ ĝ ν) τ + ξ (ê ê ν) + η [ = q τ ( 1 J ) + (ξ τ J ) ξ + ( η τ J ) η + ( ζ τ J ) ζ + (f f [ ν Re ) ( ξ y J ) ξ + ( η y J ) η + ( ζ y J ) ζ ] + (e e ν Re ) ] + (g g ν Re ) [ ( ξ x J ) ξ + ( η x J ) η + ( ζ ] x J ) ζ [ ( ξ z J ) ξ + ( η z J ) η + ( ζ z J ) ζ ]. (5) All four terms on the RHS of Eq. (5) vanish analytically. The difficulty arises when discrete representations of the temporal and spatial derivatives are used. The discrete forms of the last three terms are zero when identities of transformation metrics are written in conservative form: ξ x /J = (y η z) ζ (y ζ z) η, η x /J = (y ζ z) ξ (y ξ z) ζ, ζ x /J = (y ξ z) η (y η z) ξ,, (6) and central differences are used [19]. However, this is not true for the first term because of the mixed temporal and spatial derivatives. The first term set to zero is referred to as the geometric conservation law [19]. The most straightforward approach of accounting for this term is to simply include it in the discrete governing equation, representing the non-transformed governing equations more accurately [20], as it done in Eq. (3). Equations (2) and (3) are solved using a algorithm based on the method of artificial compressibility [21,22]. The physical time derivatives in the momentum equation, Eq. (3), are differenced using a second-order, three-point, backward-difference formula. To solve time-discretized

4 734 DU Gang and SUN Mao momentum equations for a divergence free velocity at a new time level, a pseudo-time level is introduced into the equations and a pseudo-time derivative of pressure divided by an artificial compressibility constant is introduced into the continuity equation. The resulting system of equations are iterated in pseudo-time until the divergence of the velocity approaches zero. Note that the GCL term is treated explicitly at each pseudo-time level. The derivatives of the viscous fluxes in the momentum equations are approximated using second-order central differences. For the derivatives of the convective fluxes, upwind differencing based on the flux-difference splitting technique is used. Details of this formulation for the incompressible Navier-Stokes equations can be found in Ref. [21 22]. For the computations in this paper, third-order upwind differencing was used at the interior points, and second-order upwind differencing was used at points next to the boundaries. For far field boundary conditions, at the inflow boundary, the velocity components are specified as free-stream conditions while the pressure is extrapolated from the interior; at the outflow boundary, the pressure is set to be equal to the free-stream static pressure and the velocity is extrapolated from the interior. On the airfoil surfaces, impermeable wall and noslip boundary conditions were applied, and the pressure on the boundary is obtained from the normal component of the momentum equation. 1.2 Grid deformation approach Of all the grid-deforming strategies available, trans-finite interpolation (TFI) is commonly used for aeroelastic applications. It is a simple, algebraic method to generate or update the grid at every time step. The method may consist of both connecting surfaces with straight lines in the body normal direction and preserving the arc-length distribution between nodes, or a simple, linear distribution of translational displacements. This method can accommodate simple surface deformations, but grid quality may suffer for complex geometries or moderate deformations [20]. Morton, et al. [20] developed a modified version of TFI that could solve, to some extent, the above problems. In the present study, the wing geometry is not complex, but the deformation can be moderate. Therefore, a procedure of combining the method of the modified TFI [20] and the method of solving Poisson equation is used. The procedure is outlined as follows. An initial grid is generated for the un-deformed wing by a Poisson solver that is based on the work of Hilgenstock [23]. At each time step during wing deformation, first TFI is used to give a preliminarily updated grid; then this grid is improved by using the Poisson solver. Since the TFI-updated preliminary grid is not too different from the Poisson solver-improved final grid, the solution time of using the Poisson solver is not large. By so doing, a grid of good quality at every time step is obtained. 2 Results and discussion 2.1 Test of the flow solver Before proceeding to make any observations pertaining to the physical aspect of the flow, validation of the flow solver must be conducted. The solver is tested in two ways. First, to test the flexible dynamic grid method, the solver is used to compute the flows of a wing performing flapping motion over a deforming grid and a rigid moving grid. Second, it is tested against experimental data of a model bumble bee wing in revolving motion by Usherwood and Ellingtion [7]. In the first test, flows of a model fruit fly wing (Fig. 2(a)) in flapping motion are computed in two ways: rigid grid motion corresponds to whole grid system moves with the wing; whereas in flexible grid motion, far boundary is fixed and the inner grid is deformed as the wing moves. The flapping motion is the same as that used in Ref. [24] (also see below). For the rigid grid, the dimensions are , around the wing section, in the normal direction and in the spanwise direction, respectively, and the outer boundary is located 20 chord lengths away from

5 Effects of unsteady deformation of flapping wing on its aerodynamic forces 735 the wing (as shown in previous studies, e.g., Ref. [24], this location of the outer boundary is far enough). For the deforming grid, the outer boundary is located 30 chord lengths from the wing (since the outer boundary is fixed and the surface boundary moving with the wing, distance between them is made larger to avoid severe grid twist). The number of grid points around the wing section is the same as that of the rigid grid, but the grid points in the normal and spanwise directions are increased to 90 and 120 respectively, to make the grid density the same as that of the rigid grid. Figure 3 compares potions of the rigid and deformed grids. Figure 4 compares the time courses of the aerodynamic force coefficients in one cycle between the cases of rigid moving grid and deforming grid. Results of the two solutions are almost identical, as they should be. In the second test, the set-up of Usherwood and Ellingtion [7] is followed and the aerodynamic forces are computed for the model wing in azimuthal rotation using the deforming grid (outer boundary fixed, grid deforming when the wing rotates from 0 to 120 ). In order to make comparisons with the experiment data, lift and drag coefficients are averaged between 60 and 120 from the initial start of rotation of the wing. Figure 5 compares the computed and experimental force coefficients. In the whole range of α in the experiment (from 20 to 100 ) the computed C L agrees well with the experiment values. The computed C D also agrees with the measured values except when α is larger than 60. The above cross validations have given overall confidence in the present solver with griddeformation. z y Wing root Axis of pitching rotation α (a) R O (b) x Fig. 2 (a) The wing planform used (b) Sketches of the reference frames and the wing motion Fig. 3 Comparison of a portion of the rigid grid and that of the deformed grid 2.2 The flapping motion and the wing deformation The planform of the wing used in this study is the same as that of a fruit fly wing (Fig. 2(a)). The wing section is a flat plate of 3% thickness with rounded leading and trailing edges. The

6 736 DU Gang and SUN Mao Fig. 4 Comparison of time courses of the lift (C L) and drag (C D) coefficients calculated with rigid grid and deformed grid (τ c is wingbeat period) Fig. 5 Comparison of the calculated and measured lift and drag coefficients of a wing in azimuthal rotation radius of the second moment of wing area, denoted by r 2, is 0.6R, where R is the wing length. oxyz is an inertial coordinate system (Fig. 2(b)), where the origin is at the wing base, x and y form the horizontal plane (x points backwards), and the z-direction is vertical. First, the flapping motion of the wing without deformation (rigid flat plate wing) is described. On the basis of the available data of free-flying insects [8], the flapping motion can be approximated as follows: the azimuthal rotation of the wing about the z axis (see Fig. 2(b)) which is called translation, and the pitching rotation of the wing near the end of a halfstroke (downstroke or upstroke) and at the beginning of the following half-stroke (upstroke or downstroke), which is called rotation. The velocity at r 2, due to wing translation, is called translational velocity (u t ) and is approximated by the simple harmonic function: u + t = 0.5π sin(2πτ/τ c ), (7) where u + t is the non-dimensional translational velocity, defined as u + t = u t /U (U is the mean translational velocity of the wing over a half-stroke and is used as reference velocity; related to the stroke amplitude (Φ), the stroke frequency (n) and r 2 by U = 2Φnr 2 ), τ is the nondimensional time, defined as τ = tu/c (t is the time), τ c is the non-dimensional period of a flapping cycle. Denoting the azimuthal-rotational speed as φ, we have φ(τ) = u t /r 2. The geometric angle of attack of the wing is denoted by α. It takes a constant value except at the beginning or near the end of a half-stroke. The constant value is denoted by α m, which is the angle of attack during the down- or upstroke translation; here we assume that α m in the downstroke is the same as that in the upstrokes. Around the stroke reversal, α changes with time and the angular velocity, α, is given by α + = 0.5 α + 0 {1 cos[2π(τ τ r)/ τ r ]}, τ r τ (τ r + τ r ), (8) where the non-dimensional form α + = αc/u, α + 0 is a constant, τ r is the non-dimensional time at which the rotation starts, τ r is the non-dimensional time interval over which the rotation lasts (termed rotation duration). In the time interval of τ r, the wing rotates from α = α m to α = 180 α m, Therefore, when α m and τ r are specified, α + 0 can be determined. Around

7 Effects of unsteady deformation of flapping wing on its aerodynamic forces 737 the next stroke reversal, the wing would rotate from α = 180 α m to α = α m, the sign of the RHS of Eq. (8) should be reversed. On the basis of flight data [8], τ r is approximately 0.2τ c. The data also show that, in general, the wing rotation can be taken as symmetrical rotation, i.e., half the wing rotation is conducted near the end of a half-stroke and another half at the beginning of the next half-stroke (as a result, τ r can be determined from the value of τ r ). Next, the wing deformation is described. As observed by Ellington [8] and Ennos [11], for flies, bees, moths and many other insects, in general, twist and camber change with time and reverse their direction during the stroke reversal (i.e., during wing rotation), and during the mid-portion of a half-stroke, the deformation is approximately constant. They also observed that the geometric angle of attack varied approximately linearly along the wing span (termed linear twist). On the basis of the observations, in the present study, we model the deformation as follows. The twist is assumed linear, and thus it can be determined by the twist angle (the difference in the geometrical angle of attack between wing-tip and wing-root), denoted as α tw (t). For the camber, we use equations of the mean lines of the NACA four-digit wing sections: y c = m p 2 (2px x2 ) (0 x p), m y c = (1 p) 2 [(1 2p) + 2px (9) x2 ] (p < x 1), where x is the abscissa of a point on the chord line (divided by chord length), y c the ordinate of a point on the mean line, m the maximum ordinate of mean line expressed as fraction of chord, and p the chordwise position of maximum ordinate. We assume that the maximum ordinate of the mean line is in the mid chord position, i.e., p = 0.5, and thus the camber deformation is determined by m(t). For convenience, we express time during a stroke cycle as a nondimensional time parameter ˆτ, such that ˆτ = 0 at the start of the downstroke and ˆτ = 1 at the end of the subsequent upstroke. The expressions used for α tw (t) and m(t) are as follows: { α tw,0, 0.1 < ˆτ < 0.4, α tw (t) = (10) α tw,0 [1 2(ˆτ 0.4)/0.2], 0.4 < ˆτ < 0.6; { m 0, 0.1 < ˆτ < 0.4, m(t) = (11) m 0 [1 2(ˆτ 0.4)/0.2], 0.4 < ˆτ < 0.6; where α tw,0 and m 0 are constants which specify the twist and camber in the mid portion of a half-stroke. Equations (10) and (11) give the variations of α tw (t) and m(t) during the mid portion of the downstroke (ˆτ = ) and the rotation period between the downstroke and the upstroke (ˆτ = ); the variations of α tw (t) and m(t) during the mid portion of the upstroke and the upstroke-to-downstroke rotation are similar, except that the direction of change is reversed. The above formula for deformation are based on the observation of the flight of flies, bees, moths and many other insects [8,11]. Some of the insects that are not considered in Refs. [8] and [11] may have unsteady deformation even during the mid-portion of the down or upstroke. These insects need to be considered in the future work. The lift coefficient C L is defined as F L /(0.5ρU 2 S) (F L is the lift, S is the wing area and ρ the fluid density), and the C D drag coefficient is defined similarly. In the non-dimensional Navier-Strokes equations, the only parameter that needs to be specified is the Reynolds number (Re = Uc/ν = 2Φnr 2 c/ν, where ν is the kinematic viscosity). In the equations of wing motion (Eqs. (7) and (8)), τ c and α m need to be specified. Note that τ c is related to Φ by τ c = 2Φ(r 2 /R)(R/c) (R, r 2 /R and R/c are known). Thus, in calculating the aerodynamic force coefficients of the flat plate wing (wing without deformation), Re, Φ and α m need to be specified. For the case of wing with deformation, twist and camber, i.e., α tw,0 and m 0, need to be specified additionally. In this study, a typical value of Φ ( Φ = 150 ) is used; a range of α m values (24 56 ) and two Re values (Re = 200 and Re = 4000) are considered.

8 738 DU Gang and SUN Mao 2.3 The effects of camber and twist on the aerodynamic properties of a flapping wing In order to analyze the effects of deformation, camber and twist are treated separately first, and then the effects of their combination are given. First, the effects of camber are examined. Figure 6 gives the time courses of C L and C D in one cycle for the wing with various magnitudes of camber deformation (m 0 = ) at Re = 200 and α m = 40. Results for the case of zero camber (flat plate wing; m 0 = 0) are also included for comparison (for all the computations, grid dimensions are used; tests of grid density effects have been performed and it has been shown that denser grid is not needed). It is seen that in the mid portion of the half-stroke, camber increases C L, and it has less effect on C D, but during the stroke-reversal rotation period (e.g., ˆτ = ), the camber has more effect on C D than on C L. Note that during the stroke reversal, the wing has very small velocity; the relatively large increase in C D is due to the unsteady deformation effect, i.e., although the wing is moving slowly, the camber is varying with time at a fast rate, giving the mid part of the wing chord a large speed. Fig. 6 The lift (C L) and drag (C D) coefficients versus time during one cycle for the wing with various magnitudes of camber deformation (Re = 200, α m = 40 ) The mean lift ( C L ) and drag ( C D ) coefficients are given in Table 1. When camber is less than 10%, C L increased with increasing camber; when camber is larger than 10%, C L increases very little with the increase of camber. CD is also increased by the camber, but the increment is smaller than that of C L, that is, lift-to-drag ratio is increased by the camber. For example, with m 0 = 0.1 (10% camber), C L is increased by about 15% and lift-to-drag by 12%, compared with the flat-plate wing. The result that camber deformation increases the aerodynamic forces is expected because camber can increase the asymmetry of the upper and lower surfaces of the wing (similar to the effect of the effective angle of attack being increased). Figures 7 and 8 show the vorticity contour plots at half-wing length at various times in one cycle, for the flat plate wing and the dynamically deforming wing with 10% camber, respectively. For the case of flat plate wing (Fig. 7), the results are identical to those shown in previous works [5,24] ; that is, the LEV does not shed in an entire half-stroke and the large C L and C D are due to the delayed stall mechanism. For the case of a wing with camber deformation (Fig. 8), a LEV also exists and does not shed in an entire half-stroke; the difference is that the LEV is distorted by the camber.

9 Effects of unsteady deformation of flapping wing on its aerodynamic forces 739 Table 1 The mean lift, drag coefficients and lift-to-drag ratio for the wings with various deformations (α m = 40 ) m 0 α tw,0 /( ) Re = 200 Re = 4000 C L CD CL / C D CL CD CL / C D No deformation (4%) 1.66(0%) 1.02(4%) (11%) 1.68(1%) 1.08(10%) 2.12(15%) 1.62(2%) 1.31(13%) (15%) 1.70(2%) 1.10(12%) 2.23(20%) 1.62(2%) 1.38(19%) (15%) 1.72(4%) 1.09(11%) ( 0.6%) 1.68(1%) 0.96( 2%) 1.81( 2%) 1.60(0.6%) 1.13( 2%) (10%) 1.68(1%) 1.07(9%) 2.11(14%) 1.63(2%) 1.29(11%) Note: The numbers in parentheses are percentage of increment compared to the case of no deformation. (a) τ = 0.1 (b) τ = 0.4 (c) τ = 0.5 (d) τ = 0.6 Fig. 7 The vorticity contour plots at half-wing length at various times in one cycle for the wing without deformation (Re = 200, α m = 40 ) (a) τ = 0.1 (b) τ = 0.4 (c) τ = 0.5 (d) τ = 0.6 Fig. 8 The vorticity contour plots at half-wing length at various times in one cycle for the dynamically deforming wing (m 0 = 0.10, Re = 200, α m = 40 ) Next, the effects of twist are investigated. For a linearly twisted wing, the lift is approximately zero when its geometrical angle of attack at r 2 is zero. Thus, in comparing the aerodynamic performances of a linearly twisted wing and the corresponding flat plate wing, geometrical

10 740 DU Gang and SUN Mao angle of attack at r 2 should be used. Figure 9 given the time courses of C L and C D in one cycle for wings with 20 twist at Re = 200 and α m = 40 (results of the flat plate wing are also included). The mean force coefficients are given in Table 1. The twist has very small effects on C L and C L, and it increases C D and C D slightly. Finally, the effects of combined camber and twist are examined. Figure 10 gives the time courses of C L and C D in one cycle for the deforming wing with 6% camber and 20 twist at Re = 200 and α m = 40, the corresponding C L and C D are given in Table 1. The results are rather similar to those of the corresponding cases of wing with camber deformation only. This again shows that twist has little effect on the aerodynamic forces. In the above, the effects of wing deformation on the aerodynamic forces and flows at Re = 200 and α m = 40 are investigated in detail. Flight data (e.g., Ref. [8]) show that Re for most insects range from 100 to a few thousands and α m is around 35. Here, in order to investigate the effects of Re, flows at Re = 4000 for wings with and without deformation are computed. The C L and C D results are also given in Table 1. At Re = 4000, increases in C L and in liftto-drag ratio due to the camber are larger than in the case of Re = 200 (e.g. at Re = 200, the increases in C L and C L / C D due to 10% camber are 15% and 12%, respectively; whilst at Re = 4 000, the increases are 20% and 19%, respectively). It is reasonable to expect that the effect of camber is larger for a wing with attached flow than for the wing with separated flow. For the case of high Re, the LEV is more concentrated and the vortex layer on the upper surface of the wing is thinner (see Fig. 2.3). This explains why increases in C L and C L / C D are larger for the case of higher Re. As for the effects of α m, it is apparent that the percentage of increase in C L (or C D ) due to camber deformation would be smaller for larger α m. To show this, four more values of α m (56, 48, 32 and 24 ) are considered. Table 2 gives the computed C L and C D for these cases. As expected, at larger α m, the enhancement of C L and C D due to the camber effect is smaller. Fig. 9 The lift (C L) and drag (C L) coefficients versus time during one cycle for wings with and without twist deformation (Re = 200, α m = 40 ) Fig. 10 The lift (C L) and drag (C D) coefficients versus time during one cycle for the wings with and without camber and twist deformation (Re = 200, α m = 40 )

11 Effects of unsteady deformation of flapping wing on its aerodynamic forces 741 Fig. 11 Comparison of vorticity plots at mid-stroke (ˆτ = 0.25) between the cases of Re = 200 and 4000 (m 0 = 0.06) Table 2 The mean lift, drag coefficients and lift-to-drag ratio for wings with and without deformation at various α m (Re = 200) α m/( ) No deformation α tw,0 = 20 and m 0 = 0.06 C L CD CL / C D CL CD CL / C D (19%) 0.90(10%) 1.20(8%) (15%) 1.21(4%) 1.22(10%) (10%) 1.68(1%) 1.07(9%) (11%) 2.25(2%) 0.87(9%) (8%) 2.86(1%) 0.68(7%) Note: The numbers in the parentheses are percentage of increment compared to the case of no deformation. 2.4 Some discussions on the effects of wing deformation on insect flight performance In previous works on aerodynamic force production of insect wings and on power requirements of flight in insects, a rigid flat-plate wing has been used (e.g., Refs. [25,26]). As shown above, a wing that develops camber and twist during flapping motion would produce a larger lift and lift-to-drag ratio, compared with the flat-plate wing. This means that the wing deformation could influence the maximum aerodynamic force coefficient the wing can produce and the power requirement of flight. Maximum aerodynamic force coefficient, which is important to control maneuverability, carrying loads, etc., is produced at high α m (at α m 50, see Table 2). The effect of wing deformation on aerodynamic force production is relatively small at high α m, but is still considerable. As seen in Table 2, at Re = 200 and when the wing has 6% camber and 20 twist, the maximum lift coefficient can be increased by around 8% compared to the case of rigid flat-plate wing. At higher Re and with larger deformation, the increment could be even larger. For equilibrium flight, compared to the case of rigid flat-plate wing, an insect with camber (and twist) deformation could use a smaller α m to produce a lift that supports its weight and moreover, the lift-to-drag ratio of the wing is smaller. Both these two factors will contribute to reducing the power requirement. Here, as an example, we consider the hovering flight of a bumblebee and examine how the wing deformation influences the power requirement. Similar to Ref. [26], flight data of the bumblebee are taken from Ref. [18]: mass of the insect (m) is 175 mg, wing length (R) is 13.2 mm, area of two wings (S t ) is 106 mm 2, c is 4.01 mm, r 2 /R is 0.554, n is 155 Hz, Φ is 116 and the strike plane angle is zero. On the basis of the

12 742 DU Gang and SUN Mao above data, the mean lift coefficient required for balancing the insect weight ( C L,w ) is computed as C L,w = mg/(0.5ρu 2 S t ) = 1.25 (where U = 2Φnr 2 and ρ is the fluid density), the Reynolds number is computed as Re = U c/ν = As noted above, the non-dimensional parameters required in the calculation of aerodynamic force coefficients are Re, Φ and α m. Φ and Re have been determined using the flight data. Here, α m is determined using the force balance condition. The procedure is as follows: a value of α m is guessed, the flow equations are solved and the corresponding C L is calculated, C L is compared with C L,w. If C L is not equal to C L,w, α m is adjusted. The calculation is repeated until the difference between C L and C L,w is less then After α m is determined, the power required for producing the lift that balance the insect weight is computed using the same method as that in Ref. [26]. The computed results are as follows. With the rigid flat-plate wing, α m is 25 and specific aerodynamic power (P a, aerodynamic power for unit mass) is 50 W/kg. With the deforming wing (6% camber and 20 twist based on the observed data of Ref. [8]), α m is 20 and P a is 42 W/kg. This shows that due to wing deformation, the hovering bumblebee would use a smaller wing angle of attack and save energy by about 16% compared with the case of rigid wing. 3 Conclusions Aerodynamic forces on the flapping wing are not much affected by considerable twist, but are affected by camber deformation. The effect of combined camber and twist deformation is similar to that of camber deformation. With a deformation of 6% camber and 20 twist (typical values observed for wings of many insects), lift is increased by 10 20% and lift-to-drag ratio by around 10% compared with the case of rigid flat-plate wing. As a result, the deformation can increase the maximum lift coefficient of an insect, and can decrease its power requirement of flight, e.g., for a hovering bumblebee with dynamically deforming wings (6% camber and 20 twist), aerodynamic power required is decreased by about 16% compared with the case of rigid wings. References [1] Dickinson M H, Götz K G. Unsteady aerodynamic performance of model wings at low Reynolds numbers[j]. J Exp Biol, 1993, 174(1): [2] Ellington C P, van den Berg C, Willmott A P. Leading edge vortices in insect flight[j]. Nature, 1996, 384(6610): [3] Willmott A P, Ellington C P, Thomas A R. Flow visualization and unsteady aerodynamics in the flight of the hawkmoth, manduca sexta[j]. Phil Trans R Soc Lond B, 1997, 352(1351): [4] Liu H, Ellington C P, Kawachi K, Van Den Berg C, Willmott A P. A computational fluid dynamic study of hawkmoth hovering[j]. J Exp Biol, 1998, 201(4): [5] Sun M, Tang J. Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion[j]. J Exp Biol, 2002, 205(1): [6] Dickinson M H, Lehman F O, Sane S P. Wing rotation and the aerodynamic basis of insect flight[j]. Science, 1999, 284(5422): [7] Usherwood J R, Ellington C P. The aerodynamics of revolving wings. II. Propeller force coefficients from mayfly to quail[j]. J Exp Biol, 2002, 205(11): [8] Ellington C P. The aerodynamics of hovering insect flight. III. Kinematics[J]. Phil Trans R Soc Lond B, 1984, 305(1122): [9] Ellington C P. The aerodynamics of hovering insect flight. IV. Aerodynamic mechanisms[j]. Phil Trans R Soc Lond B, 1984, 305(1122): [10] Ellington C P. The aerodynamics of hovering insect flight. VI. Lift and power requirements[j]. Phil Trans R Soc Lond B, 1984, 305(1122): [11] Ennos A R. The kinematics and aerodynamics of the free flight of some Diptera[J]. J Exp Biol, 1989, 142(1):49 85.

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