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Optimal Strouhal numbers for oscillatory propulsion in density stratified fluids

Published online by Cambridge University Press:  25 April 2025

Jiadong Wang ,
Prabal Kandel  and
Jian Deng Open the ORCID record for Jian Deng [Opens in a new window]
Jiadong Wang
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Prabal Kandel
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Jian Deng*
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
*
Corresponding author: Jian Deng, [email protected]
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Abstract

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.

JFM classification

Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1010 , 10 May 2025 , A5
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© The Author(s), 2025. Published by Cambridge University Press

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