In this work, we propose a new idea of flapping airfoil design for optimal aerodynamic performance from detailed computational investigations of flow physics. Generally, flapping motion which is combined with pitching and plunging motion of airfoil, leads to complex flow features such as leading edge separation and vortex street. As it is well known, the mechanism of thrust generation of flapping airfoil is based on inverse Karman-vortex street. This vortex street induces jet-like flow field at the rear region of trailing edge and then generates thrust. The leading edge separation vortex can also play an important role with its aerodynamic performances. The flapping airfoil introduces an alternative propulsive way instead of the current inefficient propulsive system such as a propeller in the low Reynolds number flow. Thrust coefficient and propulsive efficiency are the two major parameters in the design of flapping airfoil as propulsive system. Through numerous computations, we found the specific physical flow phenomenon which governed the aerodynamic characteristics in flapping airfoil. Based on this physical insight, we could come up with a new kind of airfoil of tadpole-shaped and more enhanced aerodynamic performance.