Unsteady Adjoint Approach for Design Optimization of Flapping Airfoils

摘要

This paper describes the work for optimizing the propulsive efficiency of flapping airfoils (i.e., improving the thrust under constraining aerodynamic work during the flapping flights by changing their shape and trajectory of motion with the unsteady discrete adjoint approach). For unsteady problems, it is essential that the time scales of motion under consideration should be properly resolved and must be compatible with the objective sought after. Both the instantaneous and time-averaged (periodic) formulations are included in this study. For the design optimization with shape parameters or motion parameters, the time-averaged objective function is found to be more useful, while the instantaneous one is more suitable for flow control. The instantaneous objective function is operationally straightforward. On the other hand, the time-averaged objective function requires additional steps in the adjoint approach; the adjoint vectors for the whole time domain must be solved in a reverse time manner. In the current study, applying the periodicity condition allows the unsteady discrete adjoint equations to be reformulated and the resulting system of equations for each sub-time-interval is solved iteratively. The design results from shape and trajectory optimizations are compared, and the physical relevance of design variables to the flapping motion at on-and off-design conditions is investigated.