The underwater Brachistochrone

TL;DR

This paper formulates and solves the underwater brachistochrone problem, accounting for buoyancy, drag, and added mass effects, revealing significant deviations from the classical cycloid and providing a tool for underwater vehicle trajectory planning.

Contribution

It introduces a comprehensive model for the underwater brachistochrone that includes all relevant fluid dynamics effects, extending classical solutions to practical underwater scenarios.

Findings

Classical cycloid becomes suboptimal near fluid density

Optimal trajectory is sensitive to Reynolds number and density ratio

Neglecting drag and added mass underestimates transit time

Abstract

The brachistochrone, the curve of fastest descent under gravity, is a cycloid when friction is absent. Underwater, however, buoyancy, viscous drag, and the added mass of entrained fluid fundamentally alter the problem. We formulate and solve the brachistochrone for a body moving through a dense fluid, incorporating all three effects together with a Reynolds-number-dependent drag coefficient. The classical cycloid becomes increasingly suboptimal as the body density approaches the fluid density, and below a critical density ratio it fails to reach the endpoint altogether. Near the critical Reynolds number for the drag crisis, the optimal trajectory is acutely sensitive to the density ratio and object size; constant-drag approximations can yield qualitatively incorrect paths. A decomposition of physical effects shows that neglecting drag and added mass together yields a predicted transit time roughly half the realised minimum, and that omitting added mass alone underestimates the transit time by approximately 20%. We extend the formulation to a three-point brachistochrone in which the trajectory must pass through an intermediate waypoint, revealing a finite reachable domain that is absent in the classical problem. The underwater brachistochrone as presented here provides a simple planning tool for short-range trajectories of buoyancy-driven underwater vehicles.