Undulatory swimming in density-stratified fluids and its scaling laws

Most existing studies of fish-like swimming have focused on homogeneous fluids, despite the widespread occurrence of density stratification in natural aquatic environments. In this work, we numerically investigate the self-propelled locomotion of an undulatory swimmer in a linearly density-stratified fluid at Reynolds numbers of upper O left parenthesis 10 cubed right parenthesis O ( 10 3 ) O(10³) . The swimmer moves freely by generating a backward-travelling body wave, allowing a direct assessment of the influence of stratification on swimming performance. Our results show that density stratification can significantly modify swimming dynamics. In particular, increasing stratification enhances the speed of horizontal swimming by amplifying the inertial reaction force on the swimmer through buoyancy restoring effects. This behaviour contrasts with previous observations that stratification suppresses vertical swimmer motion. Further analysis reveals that swimming in the stratified regime is governed by a coupling between body undulation and fluid oscillations, with the maximum swimming speed occurring when the undulation frequency approaches the buoyancy frequency. Based on this physical mechanism, we develop a reduced theoretical model that predicts optimal swimming kinematics. Since classical scaling laws derived for homogeneous fluids are no longer applicable, we propose new scaling relations for swimming speed and power expenditure in stratified fluids. We show that a multivariate power-law formulation, involving the undulatory Reynolds number and the Froude number, provides a quantitative description of swimming performance across different kinematic and stratification conditions.