Hydrodynamic interactions between mother and calf whales under optimized positioning and motion

From the onset of life, a whale calf swims in formation with its mother for not only shelter and nurturing but also for beneficial hydrodynamic interactions. However, the dynamic interactions exhibited by formations of moving solid bodies in a three-dimensional domain poses a complex and computationally intensive problem. In this study, the whales are approximated as echelon pairs of spheroids. The leading spheroid oscillates in the vertical direction under fixed parameters to mimic a mother whale's aquatic motion. The reduced size trailing spheroid, mimicking a calf whale, is optimized over multiple control parameters: position in a 3D space and amplitude and phase of bi-axial oscillatory motion. Due to the complexity by the large parametric space, an adjoint-based optimization is developed to concurrently optimize all control parameters while the computational cost remains a feasible level. This adjoint-based approach allows for in-depth understanding of the vortical structures and interactions generated between mother and calf pairs in various configurations often seen in nature, and the optimal solution may potentially go beyond the natural configurations for engineering purpose.