An adjoint-based approach to study the hydrodynamic schooling of heaving-pitching swimmers

The hydrodynamic schooling of swimmers presents fascinating flow physics with complex flows generated by individuals and their interactions. However, its further study is often hindered by the large parametric space proportionally increased with the number of individual swimmers, not to mention designing an optimal schooling for man-made underwater vehicles. An adjoint-based approach is developed here for its unique feature that the computational cost for optimization keeps almost the same with the increase of control parameters, and makes an idea choice to study hydrodynamic schooling. The preliminary study focuses on a model problem where the leading foil moves with a prescribed heaving-pitching motion, and the motion of multiple trailing foils are optimized for better hydrodynamic performance. The control parameters include the heaving amplitude, pitching amplitude, and the phase delay of each trailing foils. The optimal solution for the drag reduction of trailing foils shows that some degree of synchronization is achieved among all foils, though a small phase delay remains.