On aerodynamic modelling of an insect–like flapping wing in hover for micro air vehicles

This theoretical paper discusses recent advances in the fluid dynamics of insect and micro air vehicle (MAV) flight and considers theoretical analyses necessary for their future development. The main purpose is to propose a new conceptual framework and, within this framework, two analytic approaches to aerodynamic modelling of an insect–like flapping wing in hover in the context of MAVs. The motion involved is periodic and is composed of two half–cycles (downstroke and upstroke) which, in hover, are mirror images of each other. The downstroke begins with the wing in the uppermost and rearmost position and then sweeps forward while pitching up and plunging down. At the end of the half–cycle, the wing flips, so that the leading edge points backwards and the wing's lower surface becomes its upper side. The upstroke then follows by mirroring the downstroke kinematics and executing them in the opposite direction. Phenomenologically, the interpretation of the flow dynamics involved, and adopted here, is based on recent experimental evidence obtained by biologists from insect flight and related mechanical models. It is assumed that the flow is incompressible, has low Reynolds number and is laminar, and that two factors dominate: (i) forces generated by the bound leading–edge vortex, which models flow separation; and (ii) forces due to the attached part of the flow generated by the periodic pitching, plunging and sweeping. The first of these resembles the analogous phenomenon observed on sharp–edged delta wings and is treated as such. The second contribution is similar to the unsteady aerodynamics of attached flow on helicopter rotor blades and is interpreted accordingly. Theoretically, the fluid dynamic description is based on: (i) the superposition of the unsteady contributions of wing pitching, plunging and sweeping; and (ii) adding corrections due to the bound leading–edge vortex and wake distortion. Viscosity is accounted for indirectly by imposing the Kutta condition on the trailing edge and including the influence of the vortical structure on the leading edge. Mathematically, two analytic approaches are proposed. The first derives all the quantities of interest from the notion of circulation and leads to tractable integral equations. This is an application of the von Kármán–Sears unsteady wing theory and its nonlinear extensions due to McCune and Tavares; the latter can account for the bound leading–edge vortex and wake distortion. The second approach uses the velocity potential as the central concept and leads to relatively simple ordinary differential equations. It is a combination of two techniques: (i) unsteady aerodynamic modelling of attached flow on helicopter rotor blades; and (ii) Polhamus's leadingedge suction analogy. The first of these involves both frequency–domain (Theodorsen style) and time–domain (indicial) methods, including the effects of wing sweeping and returning wake. The second is a nonlinear correction accounting for the bound leading–edge vortex. Connections of the proposed framework with control engineering and aeroelasticity are pointed out.