Effects of wing deformation on aerodynamic forces in hovering hoverflies

We studied the effects of wing deformation on the aerodynamic forces of wings of hovering hoverflies by solving the Navier–Stokes equations on a dynamically deforming grid, employing the recently measured wing deformation data of hoverflies in free-flight. Three hoverflies were considered. By taking out the camber deformation and the spanwise twist deformation one by one and by comparing the results of the deformable wing with those of the rigid flat-plate wing (the angle of attack of the rigid flat-plate wing was equal to the local angle of attack at the radius of the second moment of wing area of the deformable wing), effects of camber deformation and spanwise twist were identified. The main results are as follows. For the hovering hoverflies considered, the time courses of the lift, drag and aerodynamic power coefficients of the deformable wing are very similar to their counterparts of the rigid flat-plate wing, although lift of the deformable wing is about 10% larger, and its aerodynamic power required about 5% less than that of the rigid flat-plate wing. The difference in lift is mainly caused by the camber deformation, and the difference in power is mainly caused by the spanwise twist. The main reason that the deformation does not have a very large effect on the aerodynamic force is that, during hovering, the wing operates at a very high angle of attack (about 50 deg) and the flow is separated, and separated flow is not very sensitive to wing deformation. Thus, as a first approximation, the deformable wing in hover flight could be modeled by a rigid flat-plate wing with its angle of attack being equal to the local angle of attack at the radius of second moment of wing area of the deformable wing. * c : mean chord length of the wing Cd : non-dimensional drag per unit wing length CD : coefficient of drag ![Graphic][1] : mean coefficient of drag Cl : non-dimensional lift per unit wing length CL : coefficient of lift ![Graphic][2] : mean coefficient of lift CP,a : coefficient of aerodynamic power ![Graphic][3] : mean coefficient of aerodynamic power LEV : leading edge vortex m : mass of the insect n : wingbeat frequency O XYZ : an inertial frame, with the XY plane in the stroke plane and O at the wing root p : pressure r : radial position along the wing length r 2 : radius of the second moment of wing area R : wing length Re : Reynolds number S : wing area ![Graphic][4] : non-dimensional time u : fluid velocity U : reference speed v : kinematic viscosity α : local angle of attack of each wing section θ : deviation angle ρ : density ϕ : stroke angle Φ : stroke amplitude ∇ : gradient operator ∇2 : Laplacian operator [1]: /embed/inline-graphic-26.gif [2]: /embed/inline-graphic-27.gif [3]: /embed/inline-graphic-28.gif [4]: /embed/inline-graphic-29.gif