Aerodynamic efficiency of flapping flight: analysis of a two-stroke model.

To seek the simplest efficient flapping wing motions and understand their relation to steady flight, a two-stroke model in the quasi-steady limit was analyzed. It was found that a family of two-stroke flapping motions have aerodynamic efficiency close to, but slightly lower than, the optimal steady flight. These two-stroke motions share two common features: the downstroke is a gliding motion and the upstroke has an angle of attack close to the optimal of the steady flight of the same wing. With the reduced number of parameters, the aerodynamic cost function in the parameter space can be visualized. This was examined for wings of different lift and drag characteristics at Reynolds numbers between 10(2) and 10(6). The iso-surfaces of the cost function have a tube-like structure, implying that the solution is insensitive to a specific direction in the parameter space. Related questions in insect flight that motivated this work are discussed.