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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 Ishihara1,*, 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


Figure 1
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  		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, 1988aGo). 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.

 

Figure 2
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  		Fig. 2. Wing torsion during the downstrokes (A) and upstrokes (B) drawn after Ennos (Ennos, 1988aGo). 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.

 

Figure 3
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  		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.

 

Figure 4
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  		Fig. 4. The relationship between the frequency ratio f/fn and the phase shift b for the damping ratios {zeta}=0.5 (red line), 1 (blue line) and 2 (black line).

 

Figure 5
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  		Fig. 5. (A) Schematic view of the analysis domain, and (B) the finite element mesh. A unit length in the z-direction is assumed.

 

Figure 6
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  		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).

 

Figure 7
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  		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).

 

Figure 8
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  		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.

 

Figure 9
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  		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.

 

Figure 10
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  		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.

 

Figure 11
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  		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).

 

Figure 12
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  		Fig. A1. The two-dimensional wing model similar to that used by Wang and colleagues (Wang et al., 2004Go) 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.

 

Figure 13
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  		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., 2004Go), respectively, while the black line represents the experimental data given by Wang and colleagues (Wang et al., 2004Go).

 

Figure 14
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  		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.

 

Figure 15
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  		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.

 

Figure 16
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  		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.

 

Figure 17
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  		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.

 

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© The Company of Biologists Ltd 2009