Unsteady aerodynamic forces of a flapping wing.

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 (Phi), mid-stroke angle of attack (alpha(m)), non-dimensional duration of wing rotation (Delta tau(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; Phi=150 degrees; alpha(m)=40 degrees; Delta tau(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 approximately 100, the lift ((L)) and drag ((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 approximately 100, (L) decreased and (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 (L) and large (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 approximately 100, lift equal to insect weight could be produced but when Re was lower than approximately 100, additional high-lift mechanisms were needed. In the range of Re above approximately 100, Phi from 90 degrees to 180 degrees and Delta tau(r) from 17% to 32% of the stroke period (symmetrical rotation), the force coefficients varied only slightly with Re, Phi and Delta tau(r). This meant that the forces were approximately proportional to the square of Phi n (n is the wingbeat frequency); thus, changing Phi and/or n could effectively control the magnitude of the total aerodynamic force. The time course of (L) (or (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).