VORTEX MERGING, OSCILLATION, AND QUASI-PERIODIC STRUCTURE IN A LINEAR-ARRAY OF ELONGATED VORTICES

Linear stability and the secondary flow pattern of the rectangular cel l flow, Psi = sin kx sin y (0 < k < infinity), are investigated in an infinitely long array of the x direction [(-infinity, infinity) x [0,p i]] 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 numeric al results of the eigenvalue problems on the infinite array show that a mode representing a global circulating vortex in the whole region (p si approximate to sin y) appears in the y-elongated cases (k > 1), whi ch confirm the secondary flow observed in Tabeling et al. [J. Fluid Me ch. 213, 511 (1990)], while a mode representing quasiperiodic arrays o f counter-rotating vortices appears in the x-elongated cases (k < 1) a t large critical Reynolds number. In large finite arrays the mode conn ected with those of the case M = infinity appears for most cases while another (oscillatory) mode appears for vortices elongated in the y di rection. The parameter region of the oscillatory modes becomes wider w hen the system size (M) becomes smaller. For a pair of counter-rotatin g vortices (M = 2) at the point k(0) between the regions of the two mo des the critical Reynolds number takes an extreme large value. Analysi s of a finite nonlinear system obtained by the Galerkin method shows t he nonlinear saturation of the critical modes, though its results are in quantitative agreement with those of the linear stability in a limi ted region of k.