Two-dimensional simulations of hydrodynamic spin coupling in a two-rotor corral
Abstract
We study hydrodynamic spin coupling in a two-rotor corral using DNS of 2D incompressible viscous fluid flow. An active rotor is driven at angular velocity W, and a nearby torque-free passive rotor selects an angular velocity w through hydrodynamic torque balance. The signed gear ratio Gamma=w/W distinguishes corotation from counterrotation, with Reynolds number Re=|Ω|r^2/ν. Motivated by a recent quasi-two-dimensional experiment, we use a DLM/FD method to compute planar phase diagrams of Γ(G,Re) at corral sizes C=3, 4.5, and 6. The planar model recovers the benchmark gap route at Re=20: an intermediate counterrotation band, a wide-gap transition to corotation, gear-ratio magnitudes of order 10^{-2}, and the observed sequence of vortex attachment, detachment, and merger. It also produces a reentrant-like gap structure with a small-gap corotation region whose relation to the experimental close-range geometric state remains unresolved. The main discrepancy is the high-Re boundary. At the experimental mid-gap transect G about 0.3, the planar gear ratio approaches zero from the counterrotating side but does not cross through Re=400; at the narrower gap G=0.22, by contrast, the planar terminal spin reverses near Re=44. Wall-traction diagnostics show that this crossing is not the experimental shear-competition mechanism: the gap-facing counterrotating arc narrows but does not collapse or deflect as in the experiment, and the reversal at G=0.22 occurs by redistribution of the integrated planar torque. The strictly planar model therefore captures the broad gap-route architecture and the existence of a Reynolds-driven spin boundary, but displaces that boundary in gap and alters its surface-stress mechanism. The remaining mismatch points to finite-depth secondary motion, end-wall stresses, and apparatus geometry as plausible contributors to the experimental shear balance.
Versions
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BibTeX
@misc{pan2026twodimensional,
title = {{Two-dimensional simulations of hydrodynamic spin coupling in a two-rotor corral}},
author = {Tsorng-Whay Pan and Jiwen He},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.01533},
eprint = {2607.01533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.01533},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-dimensional simulations of hydrodynamic spin coupling in a two-rotor corral
AU - Tsorng-Whay Pan
AU - Jiwen He
PY - 2026
DA - 2026/07/01
PB - arXiv
DO - 10.48550/arxiv.2607.01533
UR - https://arxiv.org/abs/2607.01533
AB - We study hydrodynamic spin coupling in a two-rotor corral using DNS of 2D incompressible viscous fluid flow. An active rotor is driven at angular velocity W, and a nearby torque-free passive rotor selects an angular velocity w through hydrodynamic torque balance. The signed gear ratio Gamma=w/W distinguishes corotation from counterrotation, with Reynolds number Re=|Ω|r^2/ν. Motivated by a recent quasi-two-dimensional experiment, we use a DLM/FD method to compute planar phase diagrams of Γ(G,Re) at corral sizes C=3, 4.5, and 6. The planar model recovers the benchmark gap route at Re=20: an intermediate counterrotation band, a wide-gap transition to corotation, gear-ratio magnitudes of order 10^{-2}, and the observed sequence of vortex attachment, detachment, and merger. It also produces a reentrant-like gap structure with a small-gap corotation region whose relation to the experimental close-range geometric state remains unresolved. The main discrepancy is the high-Re boundary. At the experimental mid-gap transect G about 0.3, the planar gear ratio approaches zero from the counterrotating side but does not cross through Re=400; at the narrower gap G=0.22, by contrast, the planar terminal spin reverses near Re=44. Wall-traction diagnostics show that this crossing is not the experimental shear-competition mechanism: the gap-facing counterrotating arc narrows but does not collapse or deflect as in the experiment, and the reversal at G=0.22 occurs by redistribution of the integrated planar torque. The strictly planar model therefore captures the broad gap-route architecture and the existence of a Reynolds-driven spin boundary, but displaces that boundary in gap and alters its surface-stress mechanism. The remaining mismatch points to finite-depth secondary motion, end-wall stresses, and apparatus geometry as plausible contributors to the experimental shear balance.
ER -