# Unsteady three-dimensional boundary element method for self-propelled bio-inspired locomotion

## TL;DR

This paper introduces a novel unsteady 3D boundary element method for accurately simulating bio-inspired self-propelled locomotion, enabling detailed analysis of speed, power, and efficiency in undulatory fins.

## Contribution

It combines an unsteady boundary element approach with a boundary layer solver and self-propelled equations, providing a comprehensive and validated tool for studying bio-locomotion.

## Key findings

- Validated method with experiments and analytical solutions.
- Achieved 78% propulsive efficiency for undulatory fins.
- Gained insights into wake structures and momentum jets.

## Abstract

An unsteady three-dimensional boundary element method is developed to provide fast calculations of biological and bio-inspired self-propelled locomotion. The approach uniquely combines an unsteady three-dimensional boundary element method, a boundary layer solver and self-propelled equations of motion. This novel implementation allows for the self-propelled speed, power, efficiency and economy to be accurately calculated. A Dirichlet formulation is used with a combination of constant strength source and doublet elements to represent a deforming body with a nonlinearly deforming wake. The wake elements are desingularized to numerically stabilize the evolution of the wake vorticity. Weak coupling is used in solving the equations of motion and in the boundary layer solution. The boundary layer solver models both laminar and turbulent behavior along the deforming body to estimate the total skin friction drag acting on the body. The results from the method are validated with analytical solutions, computations and experiments. Finally, a bio-inspired self-propelled undulatory fin is modeled. The computed self-propelled speeds and wake structures agree well with previous experiments. The computations go beyond the experiments to gain further insight into the propulsive efficiency for self-propelled undulating fins. It is found that the undulating fin produces a time-averaged momentum jet at 76% of the span that accelerates fluid in the streamwise direction and in turn generates thrust. Additionally, it is discovered that high amplitude motions suppress the formation of a bifurcating momentum jet and instead form a single core jet. Consequently, this maximizes the amount of streamwise momentum compared to the amount of wasted lateral momentum and leads to a propulsive efficiency of 78% during self-propelled locomotion.