Hydrodynamic torque on a steadily rotating slender cylinder

TL;DR

This study uses simulations and theory to analyze the hydrodynamic torque on a rotating slender cylinder across various aspect ratios and inertial regimes, providing empirical formulas that match numerical results.

Contribution

The paper develops an empirical model for the torque on rotating cylinders, incorporating inertial effects and finite-length corrections, validated by simulations and theoretical analysis.

Findings

Good agreement between simulations and theory for aspect ratios > 3

Wake patterns with counter-rotating vortices observed at high inertia

Empirical formulas accurately predict torque across regimes

Abstract

Using fully-resolved simulations, we investigate the torque experienced by a finite-length circular cylinder rotating steadily perpendicularly to its symmetry axis. The aspect ratio $\chi$, i.e. the ratio of the length of the cylinder to its diameter, is varied from 1 to 15. In the creeping-flow regime, we employ the slender-body theory to derive the expression of the torque up to order 4 with respect to the small parameter $1/\ln(2\chi)$. Numerical results agree well with the corresponding predictions for $\chi\gtrsim3$. We introduce an \textit{ad hoc} modification in the theoretical prediction to fit the numerical results obtained with shorter cylinders, and a second modification to account for the increase of the torque resulting from finite inertial effects. In strongly inertial regimes, a prominent wake pattern made of two pairs of counter-rotating vortices takes place. Nevertheless the flow remains stationary and exhibits two distinct symmetries, one of which implies that the contributions to the torque arising from the two cylinder ends are identical. We build separate empirical formulas for the contributions of pressure and viscous stress to the torque provided by the lateral surface and the cylinder ends. We show that, in each contribution, the dominant scaling law may be inferred from simple physical arguments. This approach eventually results in an empirical formula for the rotation-induced torque valid throughout the range of inertial regimes and aspect ratios considered in the simulations.