Two-dimensional simulations of hydrodynamic spin coupling in a two-rotor corral
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=|\Omega|r^2/\nu. Motivated by a recent quasi-two-dimensional experiment, we use a DLM/FD method to compute planar phase diagrams of $\Gamma(G,Re)$ 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.