High-efficiency swimming from hydrodynamic eigenmodes

Bioinspired underwater vehicles promise better manoeuvrability, efficiency and speed than conventional AUVs and ROVs, which are rigid and inefficient in complex, low-speed environments. The difficulty is that there is no fundamental design methodology for them. Designs are arrived at by reproducing nature through repeated trial and error, which gives no account of why a given motion works and no way to transfer it to a different body.

The question this work asks is whether the efficient motions can be derived from the hydrodynamics instead of discovered by search. If they can, bioinspired design stops being imitation and becomes something closer to engineering.

The hydrodynamics are generated with the vortex lattice method, which captures the essential flow features and the swimmer's interaction with its own wake while staying linear in the body motion. That linearity is what makes the rest possible.

The governing relationship ties the surface circulation to the displacement of the body surface. Circulation and displacement do not share dimensions, so matching them requires taking a spatial derivative of one and a temporal derivative of the other, introducing a proportionality that may vary along the span but not along the chord or in time. Assuming harmonic motion and a travelling-wave solution for the circulation turns the result into a generalised eigenvalue problem.

The circulation is then reconstructed as a weighted sum of the right eigenvectors, and the response to any prescribed body motion follows from the eigenvalues and the left eigenvectors. The method builds on Hall's original decomposition and on Smyth and colleagues, who showed in 2023 that selecting the best eigenvector can maximise efficiency even in simple inviscid models.

Dimensional analysis settles the interpretation. The eigenvalue carries units of time over length, which makes it a ratio of frequency to spatial variation: a propagation speed. Each mode therefore comes with a wave character rather than merely a shape.

Split into parts, the imaginary component corresponds to the chordwise wavenumber, which is the travelling-wave behaviour, while the real component gives the spatial growth or decay along the chord. Plotted on the complex plane the eigenvalues are not scattered: they cluster into distinct families defined by chordwise and spanwise wavenumbers. Picking an efficient motion becomes a matter of choosing the right family rather than optimising over a continuum.

The study covers three-dimensional rectangular bodies: square plates, and rectangular plates across a range of aspect ratios. Varying the aspect ratio moves the eigenvalues, so the efficient kinematics for a long thin planform are not the efficient kinematics for a square one, and the formulation says how they differ.

Not every mode is admissible. Modes that cannot contribute to propulsion, or that correspond to body movements no real vehicle could execute, are excluded on physical grounds before efficiency is considered at all.

Presented at MARINE 2025, the XI International Conference on Computational Methods in Marine Engineering, in Edinburgh. With Amanda Smyth, of the Oxford Environmental Fluid Mechanics Group, and Luca Di Mare, at the Oxford Thermofluids Institute. The full paper is in review at the Journal of Ocean Engineering and Marine Engineering.

The vortex lattice method was the right tool for establishing the result; the next stage replaces it with a higher-fidelity solver to test how far the classification survives when the physics is less forgiving.