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First published online December 16, 2008
Journal of Experimental Biology 212, 1-10 (2009)
Published by The Company of Biologists 2009
doi: 10.1242/jeb.020404

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A two-dimensional computational study on the fluid–structure interaction cause of wing pitch changes in dipteran flapping flight

Daisuke Ishihara^1,*, T. Horie¹ and Mitsunori Denda²

¹ Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 8208502, Japan
² Rutgers University, 98 Brett Road, Piscataway, NJ 08854-8058, USA

^* Author for correspondence (e-mail: ishihara{at}mse.kyutech.ac.jp)

Accepted 21 October 2008

	Summary

TOP Summary INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX References

 
In this study, the passive pitching due to wing torsional flexibility and its lift generation in dipteran flight were investigated using (a) the non-linear finite element method for the fluid–structure interaction, which analyzes the precise motions of the passive pitching of the wing interacting with the surrounding fluid flow, (b) the fluid–structure interaction similarity law, which characterizes insect flight, (c) the lumped torsional flexibility model as a simplified dipteran wing, and (d) the analytical wing model, which explains the characteristics of the passive pitching motion in the simulation. Given sinusoidal flapping with a frequency below the natural frequency of the wing torsion, the resulting passive pitching in the steady state, under fluid damping, is approximately sinusoidal with the advanced phase shift. We demonstrate that the generated lift can support the weight of some Diptera.

Key words: dipteran flight, dynamic similarity law, finite element method, fluid–structure interaction, wing torsional flexibility

	INTRODUCTION

TOP Summary INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX References

 
Flying insects depend on unsteady fluid dynamic effects to generate lift in their flapping flights. The majority of the experimental (Ellington et al., 1996; Dickinson et al., 1999; Usherwood and Ellington, 2002; Birch and Dickinson, 2003) and numerical (Liu et al., 1998; Wang, 2000; Sun and Tang, 2002; Rammamurti and Sandberg, 2002; Wang et al., 2004; Luo and Sun, 2005; Miller and Peskin, 2005) investigations have been performed with the rigid wing models that are given combined active flapping and pitching motions. These investigations have found that the unsteady fluid dynamic effects are significantly affected by the kinematical characteristics of the pitch motions such as the phase lag between pitching and flapping (Dickinson et al., 1999). On the other hand, many observations have reported the flexibility of the insect wings during flapping flights with various modes of deformation. The most significant among them is the high torsional flexibility in Diptera which is concentrated on the narrow wing basal and short root regions. The consequences of this are twofold: (1) it is hard to transmit the active torsion applied by the muscle to the outer wing and (2) the wing can twist easily to provide a passive change in the angle of attack (Ennos, 1987) or passive pitch motion.

In general, the passive pitch motion of the wings that solves the equation of motion balances three forces: wing inertial, elastic and fluid forces. According to Ennos (Ennos, 1988b), the wing inertial force accounts for much of the wing rotation at the stroke reversals in dipteran flapping flight. Although the wing inertial force might be dominant at the stroke reversals, where the acceleration is large, it is not enough to explain the passive pitch motion during the parts of the stroke other than the wing reversals. Investigation of the passive pitch motion during the entire stroke should be performed by taking into account all three of these forces. From the view point of the fluid surrounding the wings, these forces exerted on the wings act back on the fluid, producing a cycle of interaction between the fluid and the wings, termed the fluid–structure interaction (FSI), through the equilibrium on their interface.

The aim of this paper was to reveal the details of the passive pitch motion due to wing torsional flexibility and its effects on lift generation by using (a) the non-linear FSI finite element method to analyze the precise motion of the wing passive pitching and the surrounding fluid flow, (b) characterization of the insect flight using a FSI similarity law, (c) the lumped torsional flexibility model as a simplified dipteran wing and (d) the analytical wing model.

The finite element method (Ishihara and Yoshimura, 2005; Ishihara et al., 2008) was used to solve the motion of the wing model which undergoes active sinusoidal flapping motion and passive pitching in the two-dimensional fluid. The numerical simulations are guided by the FSI similarity law (D.I., M.D. and T.H., manuscript submitted), which was used to correctly incorporate data from the selected real insect, the crane fly Tipula obsolete (Ellington, 1984a; Ellington, 1984b), and the robotic fly (Wang et al., 2004) into our wing model. The lumped torsional flexibility model consists of a plate with unit extent in the z-direction and an attached spring. The former models the wing cross-section averaged over the wing span. The latter models the concentrated torsional flexibility of the wing and allows the wing to twist around its torsional axis. We also introduced the analytical wing model, a single degree of freedom mass–spring–dashpot system, to explain characteristics of the passive pitching motion in the simulation.

The passive pitching motion of our wing model simulates well the real insect's pitching motion. The model wing keeps the high attack angle during its translation and rotates at the stroke reversals. The resulting pitch angle is approximately equal to that of the selected insect, the crane fly. It is especially important that the model wing begins to twist before it changes its flapping direction. Such advanced pitch motion is necessary in order for the insect flight to increase the lift force (Dickinson et al., 1999) and is widely observed in insect flight. The analytical wing model explains the advanced pitch motion of the passive pitching as well as the torsion wave observed in the dipteran wing (Ennos, 1988b). The lift force generated by such passive pitching almost meets the force required to support the weight of the crane fly, but it is not quite enough. This could be attributed to the loosely attached leading edge vortex on the wing due to the long wing chord travel of the crane fly for the two-dimensional simulation. Indeed the lift of the fictitious insect flight, whose Reynolds number and stroke–wing chord ratio are much closer to those of the fruit fly model (Wang et al., 2004), shows a 35% increase compared with that of the crane fly flight due to the more tightly attached leading edge vortex on the wing.

	MATERIALS AND METHODS

TOP Summary INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX References

 
Non-linear finite element FSI analysis
To analyze wing motion, the surrounding fluid motion and their interaction, we used the finite element FSI analysis method (Ishihara and Yoshimura, 2005; Ishihara et al., 2008) based on the monolithic algorithm (Zhang and Hisada, 2001; Rugonyi and Bathe, 2001), which simultaneously solves the following equations.

1. Navier–Stokes equations for the incompressible viscous fluid:
   

   (1a)

   and
   

   (1b)

   where , v_i and _ij are the mass density, velocity and stress with the superscript f indicating the fluid. The fluid is assumed to be Newtonian. The stress is used, where _ij is the Kronecker delta, µ is the fluid viscosity and p is the fluid pressure. Body forces acting on the fluid are ignored for simplicity.
   

2. Equation of motion for the elastic body:
   

   (2)

   where the superscript s refers to the structural quantity. We ignore the body forces acting on the elastic body. Let us consider the equation of motion of a flapping insect wing. Assuming that the insect is hovering perfectly, the net upward aerodynamic force acting on the wing will be equal in magnitude and opposite in direction to the gravitational force on the insect. Note that the wing weight is 2–4% of the body weight in the crane fly according to Ellington (Ellington, 1984a), whose data are used in this study. Thus the gravitational force on the wing can be ignored and the body force term is dropped in Eqn 2 describing the wing motion. In this study, the second Piola–Kirchhoff stress and the Green–Lagrange strain tensors are used. The latter is defined by . We assume that the increments of these tensors are related by Hooke's law and the material non-linearity observed in biomaterials is ignored. This assumption is justified by the torsional test for the dipteran wing by Ennos (Ennos, 1988a). The test result for the wing without immobilization of its basal and root regions has shown that the relationship between the applied torque and the resulting angular displacement is approximately linear in pronation or supination.
   

3. Geometrical compatibility and equilibrium conditions on the fluid–structure interface:
   

   (3a)

   
   

   (3b)

   where and denote the outward unit normal vectors to the fluid and structure, respectively.
   

The arbitrary Lagrangian–Eulerian (ALE) method (Hughes et al., 1981) is used in order to take into account the deformable fluid–structure interface. For the elastic body, the total Lagrangian formulation is used in order to take into account the geometrical non-linearity due to the large deformation. The finite elements used for the fluid are the stabilized continuous linear velocity and pressure elements (Tezduyar et al., 1992), while those for the elastic plate are the mixed interpolation of the tensor component elements (Bathe and Dvorkin, 1985; Noguchi and Hisada, 1993). The details of the algorithm of the FSI method used in this study as well as its verification for the basic FSI problems are given by Ishihara and Yoshimura (Ishihara and Yoshimura, 2005).

We have selected the two-dimensional wing model similar to that used by Wang and colleagues (Wang et al., 2004) as a benchmark to test the validity of our method. Miller and Peskin (Miller and Peskin, 2005) have used a similar wing model to demonstrate the validity of their numerical technique. The wing is modeled by a thin rigid elastic plate of thickness h with chord length c; in our elastic model a large Young's modulus is assigned to simulate the rigid body behavior. The model is two-dimensional having a unit extent in the z-direction (out-of-plane direction) without variation in this direction. Following Wang et al. (Wang et al., 2004), an x-displacement U(t) and an angular displacement around the z-direction a(t):

(4a)

(4b)

are actively applied to the center of the wing, where A₀, β and b are the stroke amplitude, the amplitude of the pitching angle of attack and the phase shift, respectively. The common frequency f of the flapping and pitching is set to produce the Reynolds number Re=^fV_maxc/µ=75, where ^f is the fluid mass density, µ is the fluid viscosity and V_max=fA₀ is the maximum wing velocity. The details of this test are described in the Appendix, where the time histories of force coefficients given by our method agree well with the results of Wang and colleagues (Wang et al., 2004).

FSI similarity law
In this study, we introduced a dynamic similarity law for the FSI (D.I., M.D. and T.H., manuscript submitted) using the dimensional analysis of the equations of motion of the FSI system, Eqns 1a, 2 and 3b. Let U, V, L, f and P be the characteristic displacement, velocity, length, frequency and pressure, respectively. The similarity law consists of six non-dimensional numbers: Re=^fVL/µ (the Reynolds number, the ratio between the inertial force due to the convective acceleration and the viscous force), St=fL/V (the Strouhal number, the ratio between the inertial force due to the Eulerian time derivative acceleration and the inertial force due to the convective acceleration), R_s=^sL²f²/E (the ratio between the inertial force due to the Lagrangian time derivative acceleration and the elastic force), R_M=^f/^s (the mass number, the ratio between the fluid inertial force due to the Eulerian time derivative acceleration and the structural inertial force due to the Lagrangian time derivative acceleration), R_Is=fµ/E (the ratio between the fluid viscous and elastic forces) and R_Ip=^fVLf/E (the ratio between the fluid dynamic pressure and the elastic force), where the characteristic pressure P is evaluated by the magnitude of the dynamic pressure ^fV². Only four of them are independent due to the relationships R_Ip/R_Is=Re and R_M=R_sxStxR_Ip. The numbers R_s, R_Is and R_Ip are new as far as we know.

View larger version (6K): [in this window] [in a new window]   Fig. 1. Lumped torsional flexibility models. The original insect wing (A) with the arrows indicating the region of high torsional flexibility is drawn after Ennos (Ennos, 1988a). The spring model (B) shows the concept of the present lumped torsional flexibility model, and the continuum plate model (C) shows its computational implementation using the continuum plate. Note that the span-wise direction or the torsional axis is perpendicular to the plane in B and C. The pitching motion is evaluated using the angular displacement or pitch angle a, which is the slope angle of the trailing edge of the wing as shown in C. U(t), x-displacement of the wing base.

In the dimensional analysis of the equations of the motion of the FSI system, we assumed the linear strain tensor e^s_ij=1/2(u_i/x_j+u_j/x_i) and a linear relationship (Hooke's law) between the stress and strain tensors. We also used an alternative approach to the dimensional analysis of the general FSI system including, but not limited to, the linear elastic equation. We defined the following fundamental variables of the FSI system on which the system motion is dependent: L, U, V, P, ^s, ^f, E (Young's modulus) and µ. Applying the standard procedure of the dimensional analysis to these variables, we found the same non-dimensional numbers as before. The former analysis ensures that our law is the general framework of the dynamic similarity law. The latter analysis ensures that our law can be applied to the FSI system with large deformation of the elastic body.

The numerical simulations are guided by the FSI similarity law, which was used to correctly incorporate data from a selected real insect, the crane fly (Ellington, 1984a; Ellington, 1984b), and a robotic fly (Wang et al., 2004) into our wing model. It is very interesting how these parameters vary for a variety of flying insects and their flight performances change as each number is varied. In the numerical experiments of this study, comparison of the leading edge vortex behaviors between the crane fly and fictitious insect flight deals to some extent with these questions. The leading edge vortex becomes more tightly attached to the wing as Re and St are decreased while R_M and R_Is are preserved. In future work, we will tackle these interesting questions.

View larger version (5K): [in this window] [in a new window]   Fig. 2. Wing torsion during the downstrokes (A) and upstrokes (B) drawn after Ennos (Ennos, 1988a). Most of the wing torsional flexibility in Diptera is concentrated on the narrow basal and short root regions of the wings, and allows the wing to twist around its torsional axis in a span-wise direction.

Dynamic numbers:

(5a)

(5b)

(5c)

(5d)

Geometric numbers:

(6a)

(6b)

View larger version (3K): [in this window] [in a new window]   Fig. 3. A single degree of freedom mass–spring–dashpot system of the wing. U(t) and u(t), x-displacement of the wing base and center, respectively; mw, wing mass; ks, spring constant; F, force; c, chord length; M, moment.

View larger version (5K): [in this window] [in a new window]   Fig. 4. The relationship between the frequency ratio f/fn and the phase shift b for the damping ratios =0.5 (red line), 1 (blue line) and 2 (black line).

View larger version (18K): [in this window] [in a new window]   Fig. 5. (A) Schematic view of the analysis domain, and (B) the finite element mesh. A unit length in the z-direction is assumed.

Analytical wing model
In order to explain the characteristics of the passive pitching of the continuum plate model, we introduced a single degree of freedom mass–spring–dashpot system of the wing (Fig. 3) for the theoretical analysis. The wing is modeled by a rigid rod of length c and its angular displacement a(t) is assumed to be small enough that it can be measured by the relative x-displacement (t)=u(t)–U(t) of the wing center according to a(t)=(t)/(c/2) (assumption of linearization), where U(t) and u(t) are the displacement of the wing base and center, respectively. We assume that the wing mass is concentrated on the wing center and, to be consistent with the scheme of linearization, the resultant force on the wing is assumed to act on the wing center in the x-direction.

The linearized motion of the concentrated wing mass is given by:

(7)

where the first, second and third terms represent the wing inertial force f_i=m_wd²u/dt², the fluid damping force f_c=c_f(du/dt–dU/dt)= c_fd(t)/dt and the restoring force f_s=k_s'[u(t)–U(t)]=k_s'(t) due to the spring. The coefficient k_s' is the spring constant for the relative x-displacement (t)=u(t)–U(t) and is obtained by linearizing the moment and angular displacement relationship M=k_sa according to a=/(c/2) and M=f_s(c/2), with the result f_s=k_s', where:

(8)

In addition, c_f is the damping force coefficient due to the fluid and the subtraction dU/dt is the fluid advection effect induced by the wing global flapping motion.

This equation of motion is reduced to the following equation:

(9)

where F₀=m_wU₀(2f)² is the amplitude of the periodic exciting force F₀ given in terms of the amplitude U₀ (=A₀/2) and the frequency f of the spring base displacement U(t) given by Eqn 4a. The analytical solution of Eqn 9 in the steady state is given by:

(10)

with:

(11a)

(11b)

(11c)

where

(12)

and

(13)

are the natural frequency and the damping ratio, respectively.

View larger version (9K): [in this window] [in a new window]   Fig. 6. Time histories of the lift (CL, A) and drag (CD, B) coefficients for the wing motion of the flexible wing with the passive pitching (black line) and the wing motion of the rigid wing using this passive pitching as the active one (red line).

View larger version (19K): [in this window] [in a new window]   Fig. 7. Time histories of the simulated passive pitching motion (A) and the normalized lift coefficient CL generated by it (B). The pitching motion is evaluated using the angular displacement or pitch angle a, which is the slope angle of the trailing edge of the wing. The vertical lines and the attached arrows show the range of each half-stroke for the downstroke (Down) and upstroke (Up).

View larger version (5K): [in this window] [in a new window]   Fig. 8. The frame-by-frame advance of the wing motion from 7 to 8 cycles. The time interval between each frame is 0.01 cycles. The wing chord moves from right to left in the downstroke 7 to 7.5 cycles (A) and left to right in the upstroke 7.5 to 8 cycles (B). The arrows indicate the flapping direction.

View larger version (172K): [in this window] [in a new window]   Fig. 9. Fluid velocity fields from 7 to 7.9 time cycles for the crane fly (Reynolds number Re=290, Strouhal number St=0.054, corresponding to A0/c=5.9, where A0 is stroke amplitude). The time interval between each snapshot is 0.1 cycles. The arrows point in the direction of fluid velocity with their color indicating the magnitude; pink and blue correspond to Vmax=35 cm s–1 and 0, respectively. The wing chord is represented by the white line. Columns A and B represent the downstroke (from 7 to 7.4 cycles) and the upstroke (from 7.5 to 7.9 cycles), respectively. The wing moves from right to left during the downstroke and from left to right during the upstroke. The white arrows indicate the positions of the leading edge vortices. The figures in the left column show that the leading edge vortex was detached from the wing at the middle of the downstroke and then reproduced. The figures in the right column show that the leading edge vortex was loosely attached to the wing.

	RESULTS AND DISCUSSION

TOP Summary INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX References

View larger version (176K): [in this window] [in a new window]   Fig. 10. Fluid velocity fields from 4 to 4.9 time cycles for the fictitious insect (Re=200, St=0.1 corresponding to A0/c=3.2). The time interval between each snapshot is 0.1 cycles. The arrows point in the direction of the fluid velocity with their color indicating the magnitude; pink and blue correspond to Vmax=24 cm s–1 and 0, respectively. The wing chord is represented by the white line. Columns A and B represent the downstroke (from 4 to 4.4 cycles) and the upstroke (from 4.5 to 4.9 cycles), respectively. The wing moves from right to left during the downstroke and from left to right during the upstroke.

The passive pitching motion shown in Fig. 8 seems to simulate well the real insect's pitching motion. The model wing keeps the high attack angle during its translation and rotates at the stroke reversals. In addition, our wing model gives a maximum pitch angle of 50–60 deg., which lies in the range of values for the crane fly, 45–65 deg. (Ellington, 1984b). Note that the pitch angle is a non-dimensional variable. It is especially important that the model wing begins to twist before it changes its flapping direction as shown in Figs 7 and 8. Such advanced pitch motion is important for the insect flight to increase the lift force by intercepting the wake of the previous stroke (wake capture) (Dickinson et al., 1999) and is widely observed in insect flight.

View larger version (11K): [in this window] [in a new window]   Fig. 11. Time histories of the lift coefficient CL for the fictitious insect (red line) and the crane fly (black line). Note that both traces show simulations (not experimental data), and that in the fictitious insect case the leading edge vortex does not separate because of the short length of translation. The vertical lines and the attached arrows show the range of each half-stroke for the downstroke (Down) and upstroke (Up).

The analytical wing model also provides an insight into the tip to base torsion wave observed in dipteran flight (Ennos, 1988b). As shown in Fig. 4, the value of the advanced phase shift b increases as the value of f/f_n (<1) gets smaller. Since the natural frequency f_n given by Eqn 12 increases when the wing mass m_w is reduced, the phase shift is bigger for the lighter wing mass. Consequently, when we move from the base to the tip of the wing, the wing mass becomes lighter (Ennos, 1989) and the phase shift becomes larger. Such a phase shift gradient along the span-wise direction appears as the torsion wave in the three-dimensional wing. We need to be careful, however, with the limitations of our continuum plate model of the wing for which the chord length is obtained as an average over the span-wise direction and the motion represented is an average over the span. Nevertheless, our simple analytical model provides an insight into the tip to base torsion wave observed in dipteran flight (Ennos, 1988b).

Lift generated by wing motion including passive pitching
We found that the lift force generated by passive pitching almost meets the required force to support the weight of the crane fly. Fig. 7B shows the time history of the lift coefficient C_L (normalized by 0.5^fcV_max²) in the wing motion with passive pitching. Its mean value C_L,mean=0.46 gives the mean combined lift force f_L,mean=6.8x10^–5N for both wings of the crane fly, which is comparable to but smaller than the weight of the crane fly w=11x10^–5N. This could be attributed to the loosely attached leading edge vortex on the wing due to the long wing chord travel of the crane fly for the two-dimensional simulation. In the two-dimensional simulation with Re=75–115 and the stroke–wing chord ratio A₀/c=2.8–4.8 for the fruit fly, the wing reverses before the leading edge vortex has time to separate during each stroke. Thus in these cases the additional mechanism is not required to stabilize the leading edge vortex and the lift generated provides a good approximation of the corresponding lift in the three-dimensional experiment (Wang et al., 2004). In our two-dimensional simulation with Re=290 and A₀/c=5.9 (St=0.054) for the crane fly, the leading edge vortex appears to be slightly separated from the wing as shown in Fig. 9. Due to the larger wing travel length or the larger stroke–wing chord ratio and the smaller Strouhal number than those of Wang and colleagues (Wang et al., 2004), the leading edge vortex separates from the wing before the wing reverses in our two-dimensional simulation with no additional stabilization mechanism such as the span-wise flow effect. If we adopt a fictitious insect with Re=200 and A₀/c=3.2 (St=0.1), whose characteristics are closer to those of the model of Wang and colleagues (Wang et al., 2004) than to those of the crane fly, the leading edge vortex is more tightly attached to the wing as shown in Fig. 10. Consequently, the lift coefficient of the fictitious insect is higher than that of the crane fly as shown in Fig. 11 with an approximately 35% increase in the mean lift coefficient (C_L,mean=0.62) compared with that of the crane fly. Pitching in this case is also advanced, as in the case of the crane fly.

It is important to remember here that it is hard to transmit the active torsion applied by the muscle to the outer wing due to the wing torsional flexibility in Diptera (Ennos, 1987), while the wing pitch is adequately controlled to produce the lift to enable the insect to hover. Our wing model meets these criteria and explains the mechanism of insect flight with passive pitching. It seems to be natural for insects to select such a passive mechanism to minimize the effort required to move their wings. This passive mechanism might be useful as the design principle for developing micro-air vehicles, i.e. engineers can avoid the need to develop the active mechanism to drive and control the wing pitch adequately.

	APPENDIX

TOP Summary INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX References

 
Testing the validity of our FSI analysis method
We have selected the two-dimensional wing model similar to that used by Wang and colleagues (Wang et al., 2004) as a benchmark to test the capability of our method. The wing is modeled by a thin rigid elastic plate of thickness h with chord length c; in our elastic model a large Young's modulus is assigned to simulate the rigid body behavior. The model is two-dimensional having the unit extent in the z-direction (out-of-plane direction) without variation in this direction. Following Wang et al. (Wang et al., 2004), an x-displacement U(t) and an angular displacement around the z-direction a(t):

(A1a)

(A1b)

are actively applied to the center of the wing as shown in Fig. A1, where A₀=2.8c, β=/4 and b=0 are the stroke amplitude, the amplitude of the pitching angle of attack and the phase shift, respectively. The common frequency f of the flapping and pitching is set to produce the Reynolds number Re=^fV_maxc/µ=75, where ^f is the fluid mass density, µ is the fluid viscosity and V_max=fA₀ is the maximum wing velocity. The wing starts its downstroke with the initial angular displacement a(0)=a₀=0. Notice that this wing motion agrees with that used by Wang and colleagues (Wang et al., 2004) with symmetrical pitching. Miller and Peskin (Miller and Peskin, 2005) have used a similar wing model with symmetrical pitching to demonstrate the validity of their numerical technique. Fig. A2 gives the time history of lift coefficient C_L and drag coefficient C_D, normalized by 0.5^fcV_max², which shows very good agreement with the results of Wang and colleagues (Wang et al., 2004). As, in both simulations, the leading edge vortex is attached to the wing during the wing stroke, the lift coefficient agrees well with that obtained from the three-dimensional experiment even though the three-dimensional axial flow is absent.

View larger version (2K): [in this window] [in a new window]   Fig. A1. The two-dimensional wing model similar to that used by Wang and colleagues (Wang et al., 2004) as a benchmark to test the capability of our method. The x-displacement U(t) and an angular displacement around the z-direction a(t) are actively applied to the wing center.

View larger version (9K): [in this window] [in a new window]   Fig. A2. Comparisons of the time histories of CL and CD. The red and blue lines show the results given by the present method and that of Wang and colleagues (Wang et al., 2004), respectively, while the black line represents the experimental data given by Wang and colleagues (Wang et al., 2004).

View larger version (8K): [in this window] [in a new window]   Fig. A3. Time histories of CL and CD for the different positions of the axis of rotation. The black, red, blue and green lines correspond to positions O (centre), A (top), B (upper 1/10) and C (upper 1/3), respectively.

View larger version (4K): [in this window] [in a new window]   Fig. A4. The relationship between the rotational axis positions and the second peak values of the force coefficients from 3.5 to 4 cycles in Fig. A3.

View larger version (24K): [in this window] [in a new window]   Fig. A5. Convergence test where the fluid grid is refined. Mesh O (the number of nodes and elements is 3417 and 6600, respectively) is used to give the results in Figs A2, A3 and A4. The refined meshes A (the number of nodes and elements is 3905 and 7560, respectively) and B (the number of nodes and elements is 4697 and 9120, respectively) have 1.4 and 2 times finer space resolutions, respectively, around the wing.

View larger version (6K): [in this window] [in a new window]   Fig. A6. Comparison of the time histories of CL and CD. The black, blue and red lines show the results obtained using meshes O, A and B, respectively.

	Footnotes

 
D.I. is grateful for support from a Grant-in-Aid from Japan Ministry of Education, Culture, Sports, Science and Technology (No. 17760088).

	References

TOP Summary INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX References

 

Bathe, K. J. and Dvorkin, E. N. (1985). A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int. J. Numer. Methods Eng. 21,367 -383.[CrossRef]

Birch, J. M. and Dickinson, M. H. (2003). The influence of wing-wake interactions on the production of aerodynamic forces in flapping flight. J. Exp. Biol. 206,2257 -2272.[Abstract/Free Full Text]

Dickinson, M. H., Lehmann, F.-O. and Sane, P. S. (1999). Wing rotation and the aerodynamic basis of insect flight. Science 284,1954 -1960.[Abstract/Free Full Text]

Ellington, C. P. (1984a). The aerodynamics of hovering insect flight. II. Morphological parameters. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 305, 17-40.[Abstract/Free Full Text]

Ellington, C. P. (1984b). The aerodynamics of hovering insect flight. III. Kinematics. Philos. Trans. R. Soc. Lond. B, Biol. Sci. B 305,41 -78.[Abstract/Free Full Text]

Ellington, C. P., Van den Berg, C., Willmott, A. P. and Thomas, L. R. (1996). Leading-edge vortices in insect flight. Nature 384,626 -630.[CrossRef]

Ennos, A. R. (1987). A comparative study of the flight mechanism of Diptera. J. Exp. Biol. 127,355 -372.[Abstract/Free Full Text]

Ennos, A. R. (1988a). The importance of torsion in the design of insect wings. J. Exp. Biol. 140,137 -160.[Abstract/Free Full Text]

Ennos, A. R. (1988b). The inertial cause of wing rotation in Diptera. J. Exp. Biol. 140,161 -169.[Abstract/Free Full Text]

Ennos, A. R. (1989). Inertial and aerodynamic torques on the wings of Diptera in flight. J. Exp. Biol. 142,87 -95.[Abstract/Free Full Text]

Hughes, T. J. R., Liu, W. K. and Zimmerman, T. K. (1981). Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput. Meth. Appl. Mech. Eng. 29,329 -349.[CrossRef]

Ishihara, D. and Yoshimura, S. (2005). A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation. Int. J. Numer. Methods Eng. 64,167 -203.[CrossRef]

Ishihara, D., Kanei, S., Yoshimura, S. and Horie, T. (2008). Efficient parallel analysis of shell-fluid interaction problem by using monolithic method based on consistent pressure Poisson equation. J. Comput. Sci. Technol. 2, 185-196.[CrossRef]

Jensen, M. and Weis-Fogh, T. (1962). Biology and physics of locust flight. V. Strength and elasticity of locust cuticle. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 245,137 -169.[Abstract/Free Full Text]

Liu, H., Ellington, C. P., Kawachi, K., Van den Berg, C. and Willmott, A. P. (1998). A computational fluid dynamic study of hawkmoth hovering. J. Exp. Biol. 201,461 -477.[Abstract/Free Full Text]

Luo, G. and Sun, M. (2005). The effects of corrugation and wing planform on the aerodynamic force production of sweeping model insect wings. Acta Mech. Sinica. 21,531 -541.[CrossRef]

Miller, L. A. and Peskin, C. S. (2005). A computational fluid dynamics of `clap and fling' in the smallest insects. J. Exp. Biol. 208,195 -212.[Abstract/Free Full Text]

Noguchi, H. and Hisada, T. (1993). Sensitivity analysis in post-buckling problems of shell structure. Comput. Struct. 47,699 -710.[CrossRef]

Ramamurti, R. and Sandberg, W. C. (2002). A three-dimensional computational study of the aerodynamic mechanisms of insect flight. J. Exp. Biol. 205,1507 -1518.[Abstract/Free Full Text]

Rugonyi, S. and Bathe, K. J. (2001). On finite element analysis of fluid flows fully coupled with structural interactions. Comput. Model. Eng. Sci. 2, 195-212.

Sun, M. and Tang, J. (2002). Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. J. Exp. Biol. 205,55 -70.[Abstract/Free Full Text]

Tezduyar, T. E., Mital, S., Ray, S. E. and Shih, R. (1992). Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput. Methods Appl. Mech. Eng. 95,221 -242.[CrossRef]

Usherwood, J. R. and Ellington, C. P. (2002). The aerodynamics of revolving wings. I. Model hawkmoth wings. J. Exp. Biol. 205,1547 -1564.[Abstract/Free Full Text]

Wainwright, S. A., Biggs, W. D., Currey, J. D. and Gosline, J. M. (1982). Mechanical Design in Organisms. Princeton, NJ: Princeton University Press.

Wang, Z. J. (2000). Vortex shedding and frequency selection in flapping flight. J. Fluid Mech. 410,323 -341.[CrossRef]

Wang, Z. J., Birch, J. M. and Dickinson, M. J. (2004). Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computation vs robotic wing experiments. J. Exp. Biol. 207,449 -460.[Abstract/Free Full Text]

Wooton, R. J., Herbert, R. C., Young, P. G. and Evans, K. E. (2003). Approaches to the structural modeling of insect wings. Philos. Trans. R. Soc. B, Biol. Sci. 358,1517 -1587.

Zhang, Q. and Hisada, T. (2001). Analysis of fluid-structure interaction problems with structural buckling and large domain changes by ALE finite element method. Comput. Methods Appl. Mech. Eng. 190,6341 -6357.[CrossRef]

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