Modeling and Computational Fluid Dynamics Validation of a Nonholonomically Constrained Two-Rigid-Body Swimming System

Abstract

A simple nonholonomic dynamics model is developed as a low-order model for generating undulatory swim-like motions, validated through computational fluid dynamics (CFD) simulations. The rigid-body-dynamics model generates swimming motion by imposing a nonholonomic (NH) constraint on the tail of a two-body system, requiring that tail-fin velocity aligns with the tail angle, while the head moves in a straight line through a slot constraint. The system has one degree of freedom, with equations of motion derived using Lagrange multipliers. Two-dimensional CFD simulations validate the model in an incompressible Newtonian fluid, where the resolved tail fin interacts with fluid through the immersed boundary method until steady-state swimming is achieved. The validation demonstrates excellent quantitative agreement between CFD and model predictions for body orientation angle and normal fluid force across variations in fin motion amplitude, period, and Reynolds number. While an exact NH constraint point does not exist, an effective period-averaged NH location can be identified for successful model predictions. At higher Reynolds numbers, the two-body kinematics displays independence from the Reynolds number variation. The CFD data reveal that the two-body model captures the type of power-law relationship between Reynolds and Strouhal numbers governing undulatory swimming from tadpoles to whales, indicating that the simplified two-link model is representative of swimming dynamics in continuous geometries at various scales. A key limitation is that the drag force model requires a priori CFD calibration to match steady-swim velocity, limiting standalone predictive capability. The results demonstrate that the low-order NH constraint-based model effectively captures essential swimming dynamics, offering a robust alternative to existing fluid-force models.