The velocity field and instability of the flow in a rotating square duct was investigated. The velocity field was obtained for Ekman numbers less than 1/16 by combining a linear Stewartson layer solution, a nonlinear Ekman layer solution, and a local similarity assumption. The transition of the secondary-flow pattern from one pair of vortices to two pairs of vortices was studied with linear stability analysis. The resultant eigenvalue problem was solved numerically. The onset of the instability of the Ekman layer was also studied and found to be relevant to the type A and type B waves. The present theory was compared with the experimental results of Smirnov and Yurkin [Mekh. Zh. Gaza (Fluid Mechanics) 6, 24 (1983)]. The critical rotation numbers for marginal stability of both the Stewartson layer and the Ekman layer were found to be in good agreement with the measurements. The experiments of Döbner [Ph.D. dissertion, Technische Huchschule, Dormstadt (1959)] also revealed a boundary in the parameter space corresponding to the sudden change of the slopes of the drag curves, which was found at least partially related to the wavy instability of the Stewartson layer. A quasigeostrophic method was used to include the effects of Ekman friction in the instability formula. The critical rotation numbers obtained were in qualitative agreement with experimental data. Another phenomenon the experiments discovered was the pulsation of the four-vortex pattern as a certain critical Reynolds number was exceeded. This phenomenon has not yet been explained by the present analysis.
- Viscous flow,
- Taylor vortices,
- fluid mechanics
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