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

TL;DR

The paper investigates a minimal two-rigid-body swimmer constrained by a nonholonomic constraint at the tail and a slot constraint for the head. It derives the equations of motion with Lagrange multipliers, reduces to a cubic nonlinear system, and validates the low-order model against high-fidelity CFD with an immersed-boundary approach. The results show accurate predictions of body orientation and tail-force across a range of tail amplitudes, periods, and Reynolds numbers, and reveal a period-averaged effective NH location. A key limitation is that drag is calibrated to CFD to match steady swim, limiting standalone predictive capability; nonetheless, the NH-constraint framework provides a computationally efficient, scalable alternative for analyzing and designing swimming mechanisms across scales.

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.

Figures (18)

• Figure 1: Schematic of the two-rigid-body fish approximation, illustrating the coordinate system, key parameters, and constraints. The head (Body 1) and tail (Body 2) are connected by a frictionless link, with the tail's velocity constrained nonholonomically to align with its absolute angle, $\theta + \phi$.
• Figure 2: Time evolution of forward velocity for the slot-car model using parameter group 1. Each curve corresponds to a different input amplitude $a$, illustrating the effect of tail oscillation strength on steady-state propulsion. The system starts from rest and converges to a positive mean velocity with small superimposed oscillations.
• Figure 3: Steady-state orientation angle $\theta$ of the slot-car model over three oscillation periods for parameter group 1 (Table \ref{['tab:parameter_group1']}). Circles denote the full ODE simulation; solid lines represent the harmonic balance solution. As input amplitude $a$ increases, both a higher magnitude of $\theta$ and a slight phase lag become apparent. The close agreement between methods confirms that the response is dominated by the first harmonic.
• Figure 4: Steady-state constraint force $\lambda$ of the slot-car model over three oscillation periods for parameter group 1 (Table \ref{['tab:parameter_group1']}). Each curve corresponds to a different input amplitude $a$. As $a$ increases, the force magnitude grows and waveform distortions emerge, reflecting the nonlinear coupling between body kinematics and the nonholonomic constraint.
• Figure 5: Steady-state thrust in the $x$-direction, $\lambda_x = \lambda \sin(\theta + \phi)$, plotted against the tail angle $\phi$ for parameter group 1 (Table \ref{['tab:parameter_group1']}). The resulting figure-eight Lissajous curve reveals a frequency-doubling effect and minimal phase lag, consistent with thrust generation on both sides of the stroke cycle.