Optimal Strouhal numbers for oscillatory propulsion in density stratified fluids

The propulsive efficiency of flying and swimming animals propelled by oscillatory appendages typically peaks within a narrow Strouhal number range of $0.20 \lt St \lt 0.40$ . Motivated by the ubiquitous presence of stratification in natural environments, we numerically investigate the optimal Strouhal numbers $S{t_m}$ for an oscillating foil in density stratified fluids. Our results reveal that $S{t_m}$ increases with the strength of stratification characterised by the internal Froude number $Fr$ , giving rise to markedly higher values under strong stratifications compared with those observed in homogeneous fluids. The propulsive efficiency tends to maximise when there is a resonance between the oscillations of the foil and the fluid, as inferred from a fitted line in the ( $St$ , $Fr$ ) parameter space, which shows that $S{t_m}$ is proportional to $Fr^{-1}$ . We further uncover that the significant increase in $S{t_m}$ in strongly stratified regimes is fundamentally driven by fluid entrainment. During this process, the oscillating foil induces perturbations in the density field, resulting in buoyancy-driven restoring forces which alter the pressure distribution on the foil and thus the hydrodynamic forces. Notably, only under strongly stratified conditions, where dominant buoyancy effects confine the density transport to the vicinity of the oscillating foil, the intensified density perturbation due to the increase in $St$ can be effectively harnessed to enhance thrust production, thereby contributing to the elevated $S{t_m}$ . These insights suggest that oscillatory propulsors should adopt new kinematic strategies involving relatively large Strouhal numbers to achieve efficient cruising in strongly stratified environments.