The Underwater Brachistochrone

Cycloids are insufficient for underwater brachistochrone trajectories since the projectile does not reach the endpoint if it is far away, or if the projectile's density is too low. While existing work accounts for the presence of quadratic drag, we extend the analysis to the effects of buoyancy, added mass, and Reynolds number-dependent drag coefficient (C_D). We find that for shallow trajectories traversing a horizontal distance of about 4-10 times the projectile length, neglecting added mass can yield trajectories that are suboptimal by up to 20%. Furthermore, even for close-to-optimal trajectories planned without added mass, the predicted transit time can be up to 30% faster than the realised transit time. We also find that when the projectile experiences the drag crisis around Re = 2 × 10⁵, constant C_D approximations lead to suboptimal trajectories. In the most extreme cases, these trajectories may not reach the desired endpoint at all. The underwater brachistochrone as presented here has applications as a simple planning tool for short-range trajectories of underwater gliders.