Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The underwater Brachistochrone

Mohammad-Reza Alam

TL;DR

This work extends the classical brachistochrone problem to a dense fluid by incorporating gravity, buoyancy, viscous drag with a Reynolds-number dependent coefficient, and added mass through a BBO-based formulation. The resulting underwater brachistochrone deviates from the vacuum cycloid as the density ratio $\gamma = \rho_b/\rho_f$ approaches unity, and near the drag crisis ($\mathrm{Re} \approx 2 \times 10^5$) the optimal path becomes highly sensitive to $\gamma$ and object size. A three-point extension reveals a finite reachable domain after a waypoint, highlighting energy dissipation as a limiting factor in reachability absent in the classical problem. The methodology and findings provide a practical trajectory-planning tool for buoyancy-driven underwater gliders and suggest important considerations for modeling added-mass and drag in underwater optimization problems. Potential extensions include variable density schedules, non-spherical shapes, and environmental factors such as currents and stratification.

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.

The underwater Brachistochrone

TL;DR

This work extends the classical brachistochrone problem to a dense fluid by incorporating gravity, buoyancy, viscous drag with a Reynolds-number dependent coefficient, and added mass through a BBO-based formulation. The resulting underwater brachistochrone deviates from the vacuum cycloid as the density ratio approaches unity, and near the drag crisis () the optimal path becomes highly sensitive to and object size. A three-point extension reveals a finite reachable domain after a waypoint, highlighting energy dissipation as a limiting factor in reachability absent in the classical problem. The methodology and findings provide a practical trajectory-planning tool for buoyancy-driven underwater gliders and suggest important considerations for modeling added-mass and drag in underwater optimization problems. Potential extensions include variable density schedules, non-spherical shapes, and environmental factors such as currents and stratification.

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.
Paper Structure (15 sections, 15 equations, 11 figures, 1 table)

This paper contains 15 sections, 15 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Results
  4. Optimal trajectories, velocity, and drag along the path
  5. Trajectory shape.
  6. Speed and terminal velocity.
  7. Reynolds number and the drag crisis.
  8. Comparison with straight-line and cycloid paths.
  9. Improvement of the optimal path over reference trajectories
  10. Individual contributions of drag and added mass
  11. Dependence on object size and the drag crisis
  12. Transit-time map over the endpoint plane
  13. Three-point brachistochrone
  14. Reachability in the three-point problem under water
  15. Conclusions and Discussion

Figures (11)

  • Figure 1: Schematic of the underwater brachistochrone problem. An object (e.g. an underwater glider) is released at point $A$ near the water surface and must reach a target point $B$ at horizontal distance $x_e$ and depth $y_e$ in the shortest time. The dashed curve represents the optimal trajectory under the combined effects of gravity, buoyancy, drag, and added mass.
  • Figure 2: Johann Bernoulli's brachistochrone challenge as it appeared in the Acta Eruditorum, No. 6, p. 269 (June 1696). Left: cover page of the Acta Eruditorum. Center: the problem statement in Latin. Right: the accompanying figure showing points $A$ and $B$ in a vertical plane with the sought curve $AMB$. https://archive.org/details/s1id13206630/mode/2up.
  • Figure 3: Free-body diagram of an object sliding down along a trajectory $y=y(x)$ in a fluid domain.
  • Figure 4: Optimal underwater brachistochrone paths and associated flow quantities for a sphere ($R = 0.1$ m, $c_m = 1/2$) at five density ratios $\gamma$, with endpoint $(\mathtt{x}_e,\,\mathtt{y}_e) = (20,\,10)$. The classical vacuum cycloid is shown as a dashed black line. (a) Trajectories. (b) Dimensionless speed $\mathtt{v} = v/\sqrt{gL}$. (c) Reynolds number (log scale). (d) Drag coefficient $C_d$.
  • Figure 5: Effect of density ratio $\gamma$ on the underwater brachistochrone for a sphere with $R = 0.1$ m, $c_m = 1/2$, and endpoint $(\mathtt{x}_e,\,\mathtt{y}_e) = (20,\,10)$. Left: percentage improvement in transit time of the optimal path over a straight line (blue) and over the cycloid traversed in fluid (red). The improvement over the straight line peaks at approximately $27\%$ near $\gamma \approx 1.49$. For $\gamma \lesssim 1.09$, the cycloid fails to reach the endpoint. Right: dimensionless transit time $T_f$ for the optimal path (blue), straight line (dashed black), and cycloid in fluid (red).
  • ...and 6 more figures