A vortex theory of animal flight. Part 2. The forward flight of birds

Abstract

The vortex wake of a bird in steady forward flight is modelled by a chain of elliptical vortex rings, each generated by a single downstroke. The shape and inclination of each ring are determined by the downstroke geometry, and the size of each ring by the wing circulation; the momentum of the ring must overcome parasitic and profile drags and the bird's weight for the duration of a stroke period. From the equation of motion it is possible to determine exactly the kinematics of the wing-stroke for any flight velocity. This approach agrees more readily with the nature of the wing-stroke than the classical actuator disk and momentum-jet theory; it also dispenses with lift and induced drag coefficients and is not bound by the constraints of steady-state aerodynamics. The induced power is calculated as the mean rate of increase of wake kinetic energy. The remaining components of the flight power (parasite and profile) are calculated by traditional methods; there is some consideration of different representations of body parasite drag. The lift coefficient required for flight is also calculated; for virtually all birds the lift coefficient in slow flight and hovering is too large to be consistent with steady-state aerodynamics.

A bird is concerned largely to reduce its power consumption on all but the shortest flights. The model suggests that there are a number of ways in which power reduction can be achieved. These various strategies are in good agreement with observation.