Linear stability and the secondary flow pattern of the rectangular cell flow, ⌿ϭsin kx siny (0ϽkϽϱ), are investigated in an infinitely long array of the x direction ͓(Ϫϱ,ϱ)ϫ͓0,͔͔ or various finite M arrays (͓0,M /k͔ϫ͓0,͔) on the assumption of a stress-free boundary condition on the lateral walls. The numerical results of the eigenvalue problems on the infinite array show that a mode representing a global circulating vortex in the whole region (Ϸsiny) appears in the y-elongated cases (kϾ1), which confirm the secondary flow observed in Tabeling et al. ͓J. Fluid Mech. 213, 511 ͑1990͔͒, while a mode representing quasiperiodic arrays of counter-rotating vortices appears in the x-elongated cases (kϽ1) at large critical Reynolds number. In large finite arrays the mode connected with those of the case M ϭϱ appears for most cases while another ͑oscillatory͒ mode appears for vortices elongated in the y direction. The parameter region of the oscillatory modes becomes wider when the system size (M ) becomes smaller. For a pair of counter-rotating vortices (M ϭ2) at the point k 0 between the regions of the two modes the critical Reynolds number takes an extreme large value. Analysis of a finite nonlinear system obtained by the Galerkin method shows the nonlinear saturation of the critical modes, though its results are in quantitative agreement with those of the linear stability in a limited region of k.
The stability and transition of a pair of planar counter-rotating vortices in a conÿned region are investigated numerically. Direct numerical simulations in a rectangular box:where k is an aspect ratio, are done with symmetric external forcing. A pair of steady symmetric counter-rotating vortices are generated under weak forcing while they become unstable with stronger forcing. Special attention is paid to the e ect of the aspect ratio k on the stability and the transition. It is found that a steady asymmetric pattern appears for k ¿ k 0 ≈ 0:5 while an oscillating asymmetric pattern for k ¡ k 0 . In particular, the symmetric vortices become very stable around k = 0:5 (i.e. the square region). Linear stability analysis of a model ow similar to the symmetric vortices generated in the numerical simulations gives the same tendency as the above ÿndings. The results suggest that the singular property of the stability at k = k 0 ≈ 0:5 is mainly dependent on the aspect ratio k of the region but rather independent of the superÿcial di erence of the vorticity distribution.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.