Flow associated with Lighthill's elongated-body theory

TL;DR

This paper revisits and extends Lighthill's elongated-body theory to accurately predict the three-dimensional flow field around undulating fish, improving understanding of hydrodynamic forces in fish swimming.

Contribution

The paper provides a modern reinterpretation of Lighthill's theory and extends it to full 3D flow predictions, aligning well with numerical simulations.

Findings

Extended theory predicts full 3D flow fields.

Reinterpretation clarifies flow components.

Results agree with numerical simulations.

Abstract

The hydrodynamic forces acting on an undulating swimming fish consist of two components: a drag-based resistive force and a reactive force originating from the necessary acceleration of an added mass of water. Lighthill's elongated-body theory, based on potential flow, provides a framework for calculating this reactive force. By leveraging the high aspect ratio of most fish, the theory simplifies the problem into a series of independent two-dimensional slices of fluids along the fish's body, which exchange momentum with the body and neighbouring slices. Using momentum conservation arguments, Lighthill's theory predicts the total thrust generated by an undulating fish, based solely on the dimensions and kinematics of its caudal fin. However, the assumption of independent slices has led to the common misconception that the flow produced lacks a longitudinal component. In this paper, we revisit Lighthill's theory, offering a modern reinterpretation using essential singularities of potential flows. We then extend it to predict the full three-dimensional flow field induced by the fish's body motion. Our results compare favourably with numerical simulations of realistic fish geometries.