Vortex merging, oscillation, and quasiperiodic structure in a linear array of elongated vortices

Linear stability and the secondary flow pattern of the rectangular cell flow, Psi = sin kx sin y (0 < k < infinity), are investigated in an infinitely long array of the x direction [(-infinity, infinity) x [0,pi]] or various finite M arrays ([0,M pi/k] x [0,pi]) 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 (psi approximate to sin y) appears in the y-elongated cases (k > 1), which confirm the secondary flow observed in Tabeling et al. [J. Fluid Mech. 213, 511 (1990)], while a mode representing quasiperiodic arrays of counter-rotating vortices appears in the x-elongated cases (k < 1) at large critical Reynolds number. In large finite arrays the mode connected with those of the case M = infinity appears for most cases while another (oscillatory) 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 = 2) 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.