Role of fluid-structure interaction in generating the characteristic tip path of a flapping flexible wing

This study shows that characteristic modes, such as the figure-eight mode, can be created in the path of the wing tip, which is caused by the fluid-structure interaction, using a flapping model wing with two lumped flexibilities describing the elevation motion as well as the pitching motion. A direct numerical simulation based on the three-dimensional finite element method for fluid-structure interaction (FSI) analyzes the behaviors of the model wing, the surrounding air, and their interaction, where the dynamic similarity law for the FSI is used to incorporate actual insect data, and the parallel computation algorithm is used to perform the systematic parametric study. Characteristic modes, such as the figure-eight mode, are observed in the path of the wing tip from the elevation motion of the simulated wing. This motion is considered as the forced vibration caused by the interaction with the surrounding fluid excited by the flapping of the wing. Therefore, this motion can be modulated by the flexibility to change the natural frequency, which can be controlled by the muscles at the base of the wing in the actual insect. The present simulation shows that the selection between these modes in the path of the wing tip depends on the ratio between the natural frequency of the elevation motion and the flapping frequency. In the case of the figure eight, the upward elevation motion of the wing acts on the leading-edge vortex (LEV) so as to keep its momentum upon stroke reversal. Therefore, this LEV can remain in the wake of the wing after stroke reversal and enhance the next LEV. Because of this effect, the lift increases significantly as the mode of the wing tip path shifts to the figure-eight mode. This understanding will contribute to a developed field of bioinspired micro air vehicles; i.e., it will reduce the complexity of electromechanical devices that prescribe entire motions of their wings.

Figure 1

Extended lumped flexibility model, where the conventional model with lumped torsional flexibility only is extended so as to allow the wing's elevation motion from the stroke plane. The flexibility of the wing with respect to elevation from the stroke plane is modeled using the lumped flexibility at the base of the wing, which is expressed schematically as a spring. The wing base is placed at the origin O of a Cartesian coordinate system. The horizontal $x$ axis is positive in the forward direction of the longitudinal axis of the insect body. The stroke plane is set in the horizontal $xz$ plane. The flapping axis coincides with the vertical $y$ axis, and $\phi$ is the angular displacement of the flapping motion. The axis of the torsion is placed along the leading edge, and $\theta$ is the pitch angular displacement. The axis of the elevation is placed in the stroke plane and is normal to the axis of torsion, and $\chi$ is the elevation angular displacement.

Figure 2

Implementation of the lumped torsional flexibility. In the left figure (a), the torsional spring is used to illustrate the concept of the lumped torsional flexibility. In the right figure (b), it is replaced with the plate spring as its implementation. The stiff leading-edge beam and wing plate are connected by the flexible plate spring. The wing plate occupies a large part of the model wing. The plate spring works as the lumped torsional flexibility, since the plate spring is narrow.

Figure 3

Modeling of flapping motion. The time variation of $\phi$ is close to sinusoidal but has a larger acceleration or deceleration during stroke reversals and a constant velocity during the middle of each half stroke. The time variation of $\mathrm{d}\phi /\mathrm{d}t$ is expressed as a trapezoidal function, where the acceleration or deceleration time $t_{\mathrm{a}}$ is used to define the period of increasing or decreasing of the flapping velocity in each cycle, and $t_{\mathrm{a}}^{\mathrm{u}}$ and $t_{\mathrm{a}}^{\mathrm{d}}$ are for the upstroke and the downstroke, respectively. $T_{\phi }$ is the flapping period, which is the inverse of the flapping frequency $f_{\phi }$. Φ is the stroke angle.

Figure 4

Schematic view of domain decomposition.

Figure 5

Finite element mesh of the model wing. The white domain is the plate spring for the concentrated torsional flexibility, the black domain is the plate spring for the concentrated elevation flexibility, the gray domain is the stiff wing plate, and the red domain is the stiff leading edge. The chordwise lengths of the leading edge, the plate spring, and the wing plate are set as 16%, 23%, and 61%, respectively, of the total chord length.

Figure 6

Finite element mesh of the surrounding fluid.

Figure 7

Wing's motion during one stroke from 8.75 to 9.70 cycles in the case of the natural frequency of the elevation $f_{\chi }^{\mathrm{n}}=2.8f_{\phi }$. The view direction is (0, 0, 1).

Figure 8

Time histories of the wing's elevation angle $\chi$ for the various natural frerquencies of the wing's elevation motion $f_{\chi }^{\mathrm{n}}$ under the symmetric flapping condition.

Figure 9

Trajectories of the wing tip in the $\left(\phi ,\chi \right)$ plane, which show the mode of the tip path motion under the symmetric flapping condition. In each figure, the gray lines indicate each trajectory for the $n\mathrm{th}$ cycle ($n=5,...,10$), and the black line indicates the mean of them. Note that $f_{\phi }$ multiplied by the real number in the upper right-hand side of each figure denotes the value of $f_{\chi }^{\mathrm{n}}$.

Figure 10

Schematics showing the aerodynamic force acting on the wing during the middle of each half stroke. The pitch angle keeps a large angle during the middle of each half stroke, where the flapping speed is constant. Therefore, the elevation component of the dynamic pressure force acting on the wing appears as shown in this figure.

Figure 11

(a–c) are the time histories of the ideal wing's elevation motion, where the sinusoidal motions with the frequencies $2f_{\phi }$, $3f_{\phi }$, and $4f_{\phi }$ are assumed, respectively, and the unit amplitude is considered. (d–f) are the relationships between the stroke motion and the elevation motion.

Figure 12

Trajectories of the wing tip in the $\left(\phi ,\chi \right)$ plane, which show the mode of the tip path motion under the symmetric flapping condition. In each figure, the gray lines indicate each trajectory for the $n\mathrm{th}$ cycle ($n=5,...,10$), and the black line indicates the mean of them. Note that $f_{\phi }$ multiplied by the real number in the upper right-hand side of each figure denotes the value of $f_{\chi }$.

Figure 13

Relationship between $f_{\chi }^{\mathrm{n}}/\phantom{f_{\chi }^{\mathrm{n}}{f}_{\phi }}{f}_{\phi }$ and the ratio of the mean wing tip speed and the mean flapping speed in the case of the symmetric flapping.

Figure 14

Relationship between $f_{\chi }^{\mathrm{n}}/\phantom{f_{\chi }^{\mathrm{n}}{f}_{\phi }}{f}_{\phi }$ and the mean lift coefficient in the case of the symmetric flapping.

Figure 15

Time histories of the stroke angle $\phi$, the wing's elevation angle $\chi$, the pitch angle $\theta$, and the nondimensional lift $F_{\mathrm{L}}$. The black lines indicate the time histories for the $f_{\chi }^{\mathrm{n}}=2.8f_{\phi }$, and the gray lines indicate the time histories for the $f_{\chi }^{\mathrm{n}}=4.0f_{\phi }$.

Figure 16

Fluid velocity field around the wing from 9.15 to 9.25 cycles on a cylindrical plane, whose radius divided by the wing longitudinal length is approximately equal to the nondimensional radius of the second moment of the wing area, $1/3^{1/2}$. Colored arrows indicate the fluid velocity and the color indicates the magnitude from 0 (blue) to the speed three times larger than the mean flapping speed approximately (pink). The view direction is (1, 0, 1). Black bold arrows indicate the old leading-edge vortex.

Figure 17

Fluid velocity field around the wing from 9.25 to 9.35 cycles on a cylindrical plane, whose radius divided by the wing longitudinal length is approximately equal to the nondimensional radius of the second moment of the wing area, $1/3^{1/2}$. Colored arrows indicate the fluid velocity and the color indicates the magnitude from 0 (blue) to the speed three times larger than the mean flapping speed, approximately (pink). The view direction is (1, 0, 1). Black bold arrows indicate the old leading-edge vortex.

Figure 18

Fluid velocity field around the wing from 9.35 to 9.40 cycles on a cylindrical plane, whose radius divided by the wing longitudinal length is approximately equal to the nondimensional radius of the second moment of the wing area, $1/3^{1/2}$. Colored arrows indicate the fluid velocity and the color indicates the magnitude from 0 (blue) to the speed three times larger than the mean flapping speed, approximately (pink). The view direction is (1, 0, 1). Black bold arrows indicate the old leading-edge vortex.

Figure 19

Closeup view of the time histories of the pitch angle $\theta$.

Figure 20

Influence of the chordwise length of the plate spring on the results. In each trajectory of the wing tip in the $\left(\phi ,\chi \right)$ plane, the black line indicates the case using the original chordwise length of the plate spring, while the red line indicates the case using the 40% increase of the original chordwise length of the plate spring. Each trajectory is averaged for the $n\mathrm{th}$ cycle ($n=5,...,10$). $f_{\phi }$ multiplied by the real number in the upper right-hand side of each figure denotes the value of $f_{\chi }^{\mathrm{n}}$.