Hydrodynamic torque on a slender cylinder rotating perpendicularly to its symmetry axis

Abstract

Using fully-resolved simulations, we examine the torque experienced by a finite-length circular cylinder rotating steadily perpendicularly to its symmetry axis. The aspect ratio χ , 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(2χ ). Numerical results agree well with the corresponding predictions for χ 3. We introduce an 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.

Inertial settling of an arbitrarily oriented cylinder in a quiescent flow : from short-time to quasi-steady motion

“…We will distinguish two different configurations depending on the cylinder aspect ratio: moderately long rods 2 ≤ χ ≤ 30 and long fibres χ > 30. For χ ≤ 30, expressions derived through slender-body theory and direct numerical simulations by Pierson et al [18], Fintzi et al [19], Kharrouba et al…”

“…To the author's knowledge, finite-inertia effects have not been considered for a slender body rotating in a fluid at rest at infinity. Thus, the most accurate expression for this case under the Stokes flow assumption by Pierson et al [18] will be used. The expressions for F p , F q , T i r , and T Ω r are outlined in Appendix C. In both configurations, the linearized approximation is employed to describe the force as a function of the particle velocity.…”

“…Scaling analysis by Cox [5] and Pierson et al [18] has shown that this term may be small compared to the U 2 contribution when the inertia effect is not significant. However, the magnitude of this term for moderate inertia remains a topic of debate [18].…”

“…Most prior studies [15,16] have made use of the quasi-steady loads derived by Khayat and Cox [12] for slender fibers and the leading-order hydrodynamic torque resisting rotation provided by the slender body theory Batchelor [4]. Recent research has shown that the lift force and inertial torque provided by Khayat and Cox [12] and the leading-order expression for the hydrodynamic torque resisting rotation are not accurate for moderately long rods [18,19], in line with the qualitative but non-quantitative agreement reported by Cabrera et al [20] between the Khayat and Cox [12] theory and their experiments for moderately long rods. Hence, the validity of the quasi-steady assumption must be re-evaluated with accurate formulas for the loads, and the inertial correction to the loads proportional to ΩU must be considered.…”

“…These parameters include the cylinder aspect ratio, the Archimedes number which is a Reynolds number based on gravitational velocity and the density ratio. The analysis is conducted using the generalized Kirchhoff equations as proposed by Howe [6] and Mougin and Magnaudet [21], and the quasi-steady load expressions derived in Fintzi et al [19], Pierson et al [18] for moderately long rods and Khayat and Cox [12] for very elongated fibers. The proposed model is validated through scaling analysis, experimental measurements from Roy et al [16] and Cabrera et al [20] and direct numerical simulations.…”