A Minimal Swimmer Model Unveils Universal Scaling Across Reynolds Numbers

Understanding how aquatic organisms swim across vastly different scales—from microscopic bacteria to large marine mammals—remains a central challenge in biological and fluid dynamics. We present a minimal, generic swimmer model that operates without body deformation, based on time-dependent force and torque dipoles [1]. This model enables accurate 3D Navier-Stokes simulations across a vast range of Reynolds numbers (Re ∼ 10⁻⁵ to 10⁴). It reproduces essential hydrodynamic features such as propulsion and wake vortices, comparable to those observed in real fish. By introducing a new dimensionless quantity, the thrust number (Th) related to the force dipole, we identify three universal swimming regimes: Stokes (Re ∼ Th), laminar (Re ∼ Th²^/3), and turbulent (Re ∼ Th¹^/2). These scaling laws are validated against extensive experimental data spanning micro- to macro-scale swimmers. The model’s efficiency and universality make it suitable for future simulations on collective behavior in large populations. Notably, swimming performance shows a sub-linear relation between Re and Th, suggesting decreasing efficiency at high Re. Wake dynamics and vortex shedding are captured using torque dipoles, essential for inertial swimming. Our work provides a robust, scalable framework to study aquatic locomotion across biological scales.

[1] Universal Scaling Laws for a Generic Swimmer Model, B. Ventéjou, T. Métivet, A. Dupont & P. Peyla, Phys. Rev. Lett. 134, 134002 (2025)