Using adjoint-based approach to study flapping wings

Adjoint-based methods show great potential in flow control and optimization of complex problems with highor infinite-dimensional control space. It is attractive to solve an adjoint problem to understand the complex effects from multiple control parameters to a few performance indicators of the flight of birds or insects. However, the traditional approach to formulate the adjoint problem becomes either impossible or too complex when arbitrary moving boundary (e.g. flapping wings) and its perturbation is considered. Here, we use non-cylindrical calculus to define the perturbation. So that, a simple adjoint system can be derived directly in the inertial coordinate. The approach is applied to optimize the vertical motion of an oscillatory cylinder to match a pre-defined downstream velocity profile. The result quickly converges to the analytical solution.