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Fig. 3. The thin-airfoil model (A), the normal-forces model (B) and parameters used in calculation of P_ind (C). In all panels, F_R is the resultant aerodynamic force. (A) The thin-airfoil model for a wing in steady flow (e.g. Norberg, 1990), modified for non-horizontal flight. F_L is lift, and F_D is drag. β is the angle between the resultant aerodynamic force and the direction of wing movement. (B) The normal-forces model on a static wing in steady flow [after Usherwood and Ellington (Usherwood and Ellington, 2002a)]. The direction of the aerodynamic force is assumed to be normal to the surface of the wing. F_h and F_v are the horizontal and vertical forces acting on the wing, respectively. global is the angle of the wing relative to the horizontal. (C) Schematic of parameters used in calculation of P_ind. The area of the stroke plane is projected, using the angle , onto the actuator disc, which is defined as normal to F_R. _disc is the angle of the actuator disc in the global reference frame. ' is the angle between the flight velocity V and the actuator disc. As they are shown here, both _disc and ' have negative values [following Stepniewski and Keys (Stepniewski and Keys, 1984)]. V_x is the horizontal component of the flight velocity. Use of the term V_xsin_disc eliminates P_PE from the P_ind calculation.

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