On Fin Based Propulsion and Maneuvering for Uncrewed Underwater Vehicles

This paper develops a numerical framework using the WaterLily solver to investigate the hydrodynamic mechanisms of bio-inspired oscillating fin propulsion, exploring how kinematic parameters, fin flexibility, and multi-fin interactions can be optimized via Bayesian optimization to enhance thrust and maneuverability in uncrewed underwater vehicles.

The Gist

Imagine you are designing a tiny, high-tech robot to explore the deep ocean. Most underwater vehicles use propellers—spinning blades that work like a fan. But propellers can be loud, inefficient, and bulky.

This dissertation explores a different way to move: The "Fish Method." Instead of spinning, what if the robot used flapping fins, just like a tuna or a dolphin? This paper investigates how to make those fins work together to create a "super-propulsion" system.

Here is the breakdown of the research using everyday analogies:

1. The Single Fin: The "Oar in the Water"

First, the researcher looked at one single fin. Think of a person using a single paddle to move a canoe.

• The Discovery: If you just move the paddle up and down (heaving), you get some thrust. If you twist it (pitching), you get more. But the "magic" happens when you do both at the same time.
• The "Springy" Secret: The researcher found that if the fin isn't stiff like a piece of wood, but instead has a little bit of "give" (like a flexible leaf), it can actually move more efficiently. It’s like how a swimmer’s hand naturally adjusts its angle to catch the most water.

2. Maneuvering: The "Asymmetric Twist"

How do you turn a robot that only flaps? If you flap perfectly symmetrically, you just go straight.

• The Analogy: Imagine you are running through a pool. If you move your arms perfectly evenly, you go straight. But if you suddenly "cheat" and move one arm faster than the other, or tilt your body to one side, you’ll start to veer.
• The Result: The researcher proved that by "breaking the rules" of symmetry—making one stroke faster than the other or tilting the fin—the robot can steer itself without needing a separate rudder.

3. Multi-Fin Systems: "Surfing the Wake"

This is the most exciting part of the paper. What happens if you put three fins in a row, one after the other?

• The Analogy: Imagine a group of cyclists riding in a line. The person in front breaks the wind, creating a "slipstream" that makes it much easier for the people behind them to pedal.
• The "Vortex" Wave: As a fin flaps, it leaves behind little swirling whirlpools called vortices. If the second fin is timed perfectly, it doesn't just swim in "dirty" water; it actually "catches" the whirlpool left by the first fin. It’s like a surfer catching a wave—the wave provides extra energy, pushing the surfer forward with less effort.
• The Timing is Everything: If the second fin is too close or too far, or if its timing is off, it hits the whirlpool at the wrong moment and actually loses speed. It’s like trying to jump onto a moving merry-go-round; if you time it right, you hop on easily. If you time it wrong, you crash.

4. Bayesian Optimization: The "Smart Scout"

With multiple fins, there are millions of possible combinations of timing and spacing. A human could never test them all.

• The Analogy: Imagine you are looking for the best spot to camp in a massive forest. You could walk every single inch (which takes forever), or you could use a "Smart Scout." The Scout explores a few spots, notices that "the ground is getting softer and more level toward the North," and then focuses all its energy on searching that specific area.
• The Result: The researcher used a mathematical "Smart Scout" (called Bayesian Optimization) to find the perfect "sweet spot" for the fins to flap in a fraction of the time.

The Big Picture

The goal of this research is to move away from clunky, spinning propellers and toward coordinated, biological-style movement. By understanding the "dance" of the whirlpools left behind by each fin, we can build underwater robots that are faster, more efficient, and much better at maneuvering through the complex environments of our oceans.

Technical Summary

Technical Summary: On Fin-Based Propulsion and Maneuvering for Uncrewed Underwater Vehicles

Author: Parker Thomas Grobe
Date: Spring 2026
Subject: Mechanical Engineering / Fluid Dynamics

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1. Problem Statement

Traditional marine propulsion relies on continuous rotation (propellers), which necessitates separate, complex auxiliary systems (rudders, planes, thrusters) for maneuvering. This increases the cost, complexity, and drag of Uncrewed Underwater Vehicles (UUVs). In contrast, biological swimmers (e.g., tuna, dolphins) use oscillatory propulsion—periodic pitching and heaving of fins—to achieve high thrust, efficiency, and integrated maneuvering authority. The fundamental challenge addressed in this dissertation is to understand the hydrodynamic mechanisms of these oscillations and to determine how multiple fins can be coordinated to maximize propulsion and maneuverability through beneficial vortex interactions.

2. Methodology

The research utilizes a computational approach to explore the fluid-structure interaction of oscillating fins:

• Numerical Solver: The study employs WaterLily, an open-source, two-dimensional viscous incompressible Navier-Stokes solver written in Julia. It utilizes the Boundary Data Immersion Method (BDIM) to resolve fluid-structure interactions on a fixed Cartesian grid, avoiding the need for expensive dynamic re-meshing.
• Turbulence Modeling: The solver uses an Implicit Large Eddy Simulation (iLES) framework, resolving large-scale coherent vortex structures.
• Geometry & Kinematics: The primary model is a NACA 0020 airfoil. Kinematics are defined by prescribed sinusoidal heave ($h$) and pitch ($\theta$) motions, non-dimensionalized via the Strouhal number ($St$) and reduced frequency ($f^*$).
• Optimization: For complex multi-fin systems where parameter spaces (phase and spacing) become too large for exhaustive mapping, the author implements a Bayesian Optimization framework using Gaussian process surrogate models to identify high-performance configurations efficiently.

3. Key Contributions

The dissertation progresses from single-fin physics to complex multi-fin optimization:

• Single-Fin Mechanics: Established the distinction between lift-based thrust (from heave/effective angle of attack) and added-mass thrust (from rotational acceleration).
• Maneuvering Theory: Developed scaling laws to show how breaking kinematic symmetry (via pitch bias, asymmetric heave waveforms, or asymmetric stiffness) generates controlled lateral forces ($C_L$) for maneuvering.
• Vortex Interaction Heuristics: Identified that downstream fins can "extract" energy from the wake of upstream fins. The author introduced a non-dimensional phase modifier ($\phi_0$) to account for the convective travel time of vortices between fins.
• Sequential Optimization Strategy: Proved that because upstream fins are largely unaffected by downstream ones, multi-fin arrays can be optimized sequentially (adding one fin at a time) rather than through exhaustive N-dimensional searches.

4. Key Results

• Single Fin: A rigid NACA 0020 fin at $St = 0.36$ achieved $C_T = 0.47$ and $\eta = 0.37$. Introducing a leading-edge torsional spring with time-varying stiffness improved thrust by 8% and efficiency by 6.4% by allowing passive pitch to optimize the angle of attack.
• Maneuvering:
  • Pitch Bias: Lateral force $C_L$ scales linearly with imposed pitch bias ($\theta_{pb}$).
  • Asymmetric Heave: Skewing the heave waveform (fast upstroke/slow downstroke) generates lateral force, which scales with the ratio of peak heave speed and acceleration.
  • Stiffness Asymmetry: Varying spring stiffness between half-cycles produces predictable lateral forces, following a derived scaling law.
• Multi-Fin Systems:
  • Two-Fin: Optimal thrust occurs when the downstream fin encounters the upstream vortex at a specific phase ($\phi_{1,2} \approx 2\pi/3$). The phase modifier successfully collapsed data across different spacings.
  • Three-Fin: Confirmed a "hierarchy of interactions" where fin 2 is governed by fin 1, and fin 3 is governed by fin 2.
  • N-Fin Optimization: Bayesian optimization revealed that thrust-optimized systems tend toward longer wavelength traveling waves, while efficiency-optimized systems favor shorter wavelength patterns. Peak efficiency for N-fin systems was found around $N=3$ to $4$.

5. Significance

This work provides a theoretical and computational architecture for designing the next generation of bio-inspired UUVs. By demonstrating that propulsion and maneuvering can be integrated into a single oscillating surface, the research suggests a path toward vehicles with lower drag, reduced mechanical complexity, and superior control authority. The development of the phase modifier and the sequential optimization framework offers practical engineering tools to scale these systems from simple two-fin setups to complex, multi-fin arrays.