Flapping Wing Flight Can Save Aerodynamic Power Compared to Steady Flight

Flapping flight is more maneuverable than steady flight. It is debated whether this advantage is necessarily accompanied by a trade-off in the flight efficiency. Here we ask if any flapping motion exists that is aerodynamically more efficient than the optimal steady motion. We solve the Navier-Stokes equation governing the fluid dynamics around a 2D flapping wing, and determine the minimal aerodynamic power needed to support a specified weight. While most flapping wing motions are more costly than the optimal steady wing motion, we find that optimized flapping wing motions can save up to 27% of the aerodynamic power required by the optimal steady flight. We explain the cause of this energetic advantage.

Figure 1

Optimization of fixed wing kinematics for a wing of elliptical cross section of aspect ratio $1/4$ and chord length $6.8\times {10}^{-2}\mathrm{cm}$. (a) Vertical force $F_y$ as a function of the distance traveled at the optimal angle of attack. In the inset, vorticity field around the wing at steady state. (b) Specific power as a function of the angle of attack $\alpha$. For each $\alpha$, the velocity is chosen to support a weight of 0.5 mg at steady state. In the range of between 20° and 40°, containing the optimal angle of attack, the flow around the wing is separated, exhibiting dynamic stall followed by periodic vortex shedding. The minimum specific power $P_s=0.245$ occurs at a velocity of $2.94\mathrm{m}/\mathrm{s}$ and an angle of attack of 27.5°. Each data point is obtained by averaging several periods after the flow has reached a periodic state. The numerical convergence of each simulation is checked by comparing computations using a $128\times 256$ grid and a $256\times 512$ grid.

Figure 2

(a) Optimized flapping motion. The optimal flapping motion found here uses 27% less power than the optimal fixed wing kinematics. The aerodynamic power is calculated after steady state is reached by averaging over four periods of motion. (b) Sinusoidal motion [4]. Wing chords [gray (red)] and the aerodynamic force (black) are shown equally spaced in time. The optimized flapping motion in (a) requires only 38% of the aerodynamic power used by the sinusoidal motion in (b) to lift the same weight. We set the amplitude to be the arclength traveled by the wing of a typical fruit fly at $2/3$ of its length, which is about 6 chords, and the profile of the wing velocity to be almost square ($K=0.9$) as previously found by a quasisteady analysis [5]. A typical fruit fly has a mass $m\approx 1\mathrm{mg}$. Its wing has a radius $r\approx 0.2\mathrm{cm}$, a mean chord $\overline{c}\approx 6.8\times {10}^{-2}\mathrm{cm}$, and a typical flapping frequency of 250 Hz [18, 19]. Thus, each wing supports a weight of about 0.5 mg flapping at a Reynolds number of about 100.

Figure 3

Fluid force ($F_x$ and $F_y$) and vorticity field ($\omega$) during one period of optimized wing motion.

Figure 4

Wing-wake interaction near wing reversal. (a) Velocity field immediately after wing reversal in the unperturbed case. (b) Velocity field after the leading edge of the previous stroke has been artificially removed. (c),(d) Vertical force and power for half a period of motion. The traces in solid (blue) lines and dashed (red) lines correspond to the force and power generated by the unperturbed flow in (a) and to the modified flow in (b), respectively. The interaction with the leading edge vortex of the previous stroke increases both the lift and fluid power. However, the net effect is beneficial to the efficiency of lift production, as the lift in the shaded region increases threefold while the required power only increases by 50%.