Unsteady aerodynamic forces of a flapping wing

Abstract

The unsteady aerodynamic forces of a model fruit fly wing in flapping motion were investigated by numerically solving the Navier–Stokes equations. The flapping motion consisted of translation and rotation [the translation velocity (u_t) varied according to the simple harmonic function (SHF), and the rotation was confined to a short period around stroke reversal]. First, it was shown that for a wing of given geometry with u_t varying as the SHF, the aerodynamic force coefficients depended only on five non-dimensional parameters, i.e. Reynolds number (Re), stroke amplitude (Φ), mid-stroke angle of attack (α_m), non-dimensional duration of wing rotation (Δτ_r) and rotation timing [the mean translation velocity at radius of the second moment of wing area (U), the mean chord length (c) and c/U were used as reference velocity, length and time, respectively]. Next, the force coefficients were investigated for a case in which typical values of these parameters were used (Re=200;Φ =150°; α_m=40°; Δτ_r was 20% of wingbeat period; rotation was symmetrical). Finally, the effects of varying these parameters on the force coefficients were investigated. In the Re range considered (20–1800), when Re was above ∼100, the lift (C̄_L) and drag (C̄_D) coefficients were large and varied only slightly with Re (in agreement with results previously published for revolving wings); the large force coefficients were mainly due to the delayed stall mechanism. However, when Re was below∼ 100, C̄_L decreased and C̄_D increased greatly. At such low Re, similar to the case of higher Re, the leading edge vortex existed and attached to the wing in the translatory phase of a half-stroke; but it was very weak and its vorticity rather diffused, resulting in the small C̄_L and large C̄_D. Comparison of the calculated results with available hovering flight data in eight species (Re ranging from 13 to 1500) showed that when Re was above∼ 100, lift equal to insect weight could be produced but when Re was lower than ∼100, additional high-lift mechanisms were needed. In the range of Re above ∼100, Φ from 90° to 180° and Δτ_r from 17% to 32% of the stroke period (symmetrical rotation), the force coefficients varied only slightly with Re, Φ and Δτ_r. This meant that the forces were approximately proportional to the square of Φn (n is the wingbeat frequency); thus, changing Φ and/or n could effectively control the magnitude of the total aerodynamic force. The time course of C̄_L (or C̄_D) in a half-stroke for u_t varying according to the SHF resembled a half sine-wave. It was considerably different from that published previously for u_t, varying according to a trapezoidal function (TF) with large accelerations at stroke reversal, which was characterized by large peaks at the beginning and near the end of the half-stroke. However, the mean force coefficients and the mechanical power were not so different between these two cases (e.g. the mean force coefficients for u_t varying as the TF were approximately 10% smaller than those for u_t varying as the SHF except when wing rotation is delayed).