Wing kinematic and aerodynamic compensations for unilateral wing damage in a small phorid fly

To investigate the way in which very small insects compensate for unilateral wing damage, we measured the wing kinematics of a very small insect, a phorid fly (Megaselia scalaris), with $16.7\%$ wing area loss in the outer part of the left wing and a normal counterpart, and we computed the aerodynamic forces and power expenditures of the phorid flies. Our major findings are the following. The phorid fly compensates for unilateral wing damage by increasing the stroke amplitude and the deviation angle of the damaged wing (the large deviation angle gives the wing a deep U-shaped wing path), unlike the medium and large insects studied previously, which compensate for the unilateral wing damage mainly by increasing the stroke amplitude of the damaged wing. The increased stroke amplitude and the deep U-shaped wing path give the damaged wing a larger wing velocity during its flapping motion and a rapid downward acceleration in the beginning of the upstroke, which enable the damaged wing to generate the required vertical force for weight support. However, the larger wing velocity of the damaged wing also generates larger horizontal and side forces, increasing the resultant aerodynamic force of the damaged wing. Due to the larger aerodynamic force and the smaller wing area, the wing loading of the damaged wing is 25% larger than that of the wings of the normal phorid fly; this may greatly shorten the life of the damaged wing. Furthermore, because the damaged wing has much larger angular velocity and produces larger aerodynamic moment compared with the intact wing of the damaged phorid fly, the aerodynamic power consumed by the damaged wing is $38\%$ larger than that by the intact wing, i.e., the energy distribution between the damaged and intact wings is highly asymmetrical; this may greatly increase the muscle wastage of the damaged side.

Figure 1

Wing planforms of the phorid fly with a damaged left wing. $R^{\prime }$, wing length of the damaged left wing; $R$, wing length of the intact right wing.

Figure 2

Sketch showing the experimental measuring apparatus.

Figure 3

Kinematic definition: (a) coordinate system ($X$, $Y$, $Z$) with origin at the wing root and Euler angles defining the wing kinematics; (b) definition of the spanwise bending.

Figure 4

(a) Portions of the computational grid system of the wing-damaged phorid fly. (b) Time courses of the force coefficients and contour plots of nondimensional spanwise component of vorticity at the span location $r_2$ for the right wing of the normal phorid fly (solid and broken lines indicate positive and negative vorticity, respectively; the magnitude of the nondimensional vorticity at the outer contour is 2, and the contour interval is 2), calculated with three grid systems.

Figure 5

Video sequences of MSn and MSd.

Figure 6

Measured Euler angles and the maximum bending displacements of (a)–(d) the left wings and (a*)–(b*) right wings of MSn and MSd (mean ± s.d.; $n=8$ wingbeats).

Figure 7

Stroke diagrams show the wing motions of (a) MSn and (b) MSd. The curves indicate the wing paths at $0.58R$ from the wing roots; black dots on the bodies define the wing-root location; black lines in side view indicate the orientation of the wing at various times in one stroke cycle, with dots marking the leading edge; colored arrows in side view represent the velocities of the wing at $0.58R$ from the wing roots; colored arrows in top view indicate the flapping directions.

Figure 8

Diagram used to define the forces and moments. $F_x$, $F_y$, and $F_z$ denote the $x$, $y$, and $z$ components of the aerodynamic force of a wing, respectively; $M_x$, $M_y$, and $M_z$ denote the $x$, $y$, and $z$ components of the moment of a wing about the center of mass, respectively; $V$ is the total vertical force of the two wings. The direction of the axes in coordinate system $(x,y,z)$ is the same as that in coordinate system ($X$, $Y$, $Z$).

Figure 9

Time courses of the computed forces (a) $F_y$, (b) $F_x$, and (c) $F_z$ of the damaged left wing and the intact right wing of MSd.

Figure 10

Time courses of the wing velocities at $0.58R$ from the wing roots.

Figure 11

Contour of nondimensional spanwise component of vorticity of the two wings in the section at $0.58R$ from the wing roots. The vorticity is nondimensionalized by $U/c$; the counterclockwise and clockwise vorticity are denoted as solid and dashed lines, respectively; the magnitude of the nondimensional vorticity at the outer contour is 2, and the contour interval is 2.

Figure 12

Isovorticity surface plots and pressure distributions at section $0.58R$ for MSd during one stroke cycle. The vorticity is nondimensionalized by $U/c$ and the magnitude of the nondimensional vorticity is 2. TV denotes tip vortex, RV denotes root vortex, and SV denotes starting vortex. The nondimensional pressure, $C_P$, is defined as $(p-p_{\infty })/0.5\rho U^2$, where $\rho$ is the fluid density.

Figure 13

Time courses of the vertical component of the nondimensional total first moment of vorticity $\left({\gamma }_y\right)$, nondimensionalized by UcS, for the damaged left wing (a) and the intact right wing (b) of MSd, and the corresponding vertical forces (c) and (d).

Figure 14

Contributions of lift and drag to the vertical force of the left and right wings of MSd in one stroke cycle. The orange background indicates where the drag has a larger contribution to the vertical force for the damaged wing, and purple indicates where the lift has a larger contribution.

Figure 15

Respective effect of changes in the deviation and the stroke amplitude on the vertical force of the damaged wing.

Figure 16

Time courses of the computed moments (a) $M_x$, (b) $M_y$, and (c) $M_z$ of the damaged left wing and the intact right wing of MSd.

Figure 17

Time courses of the resultant aerodynamic force of (a) the left wings and (b) the right wings of MSn and MSd.

Figure 18

Time courses of the mechanical power $P$, aerodynamic power $P_a$, and inertial power $P_i$ of the right wing of MSn.

Figure 19

Time courses of the aerodynamic power of (a) the left wings and (b) the right wings of MSn and MSd.

Figure 20

Time course of $M_P$, the component of ${\mathit{M}}_{\mathbf{a}}$ in the direction of $\mathrm{\Omega }$, of the damaged left wing compared with that of the intact right wing.