Quasi-three-dimensional modeling of the normal flapping flight of an elastic membrane ornithopter

A theoretical-experimental study of the aerodynamics of a membrane ornithopter flapping its wings in the mode of normal flapping flight with flow separation from the leading edges of the wings in still air was carried out using the improved method of discrete vortices generalized for viscous vortical media. The results of high-speed video recording of ornithopter wing flaps are used in the numerical modeling. The solution of the three-dimensional problem is reduced to two-dimensional solutions for the sections of an ornithopter wing: a quasi-three-dimensional problem of the dynamics of an incompressible viscous medium was considered. The Lamb--Oseen vortex, a fundamental solution of the generalized Helmholtz equation, was used as a partial (basic) solution for each two-dimensional problem. In doing so, no restrictions were imposed on the Reynolds and Strouhal numbers and the angles of attack. The contribution of the normal force components on the ornithopter wings was investigated. Among all the components of the normal force, the inertial component dominates. Its contribution may exceed 100%. The nature of aerodynamic forces on the wings of an elastic membrane ornithopter depends on the instantaneous attached mass: it is determined by the circulation of air acceleration along the contour adjacent to the ornithopter wing. The contributions of the other components, the circulation and vortical ones, are small and negative, despite the intense vortex formation near the wing. A new kinematic parameter is proposed, namely, the normal acceleration of the trailing edge of the wing section, taken with an opposite sign. It was found that to maximize the coefficient of vertical force, this parameter should be synchronized with the pitching angle of the corresponding section of the wing. The proposed methodology provides a successful solution to three-dimensional problems by finding the aerodynamic characteristics of thin elastic wings flapping in the mode of normal flapping flight. A comparison of the calculation and experimental results shows an accuracy of about 14% for the aerodynamic force coefficients averaged over the flaps' period at the arbitrary angles of attack and the Strouhal and Reynolds numbers.