LARGE-SCALE AND PERIODIC-MODES OF RECTANGULAR CELL FLOW

Linear stability of the rectangular cell flow: Ψ=cos kx cos y (0<k<1), is studied, both numerically and analytically. Owing to its spatial periodicity, the disturbances are characterized by the Floquet exponents (α,β). Based on numerical results, it is found that two types of the critical modes with vanishingly small exponents exist. One type (large-scale mode) has an almost uniform spatial structure. The other type (periodic mode) has a structure with the same periodicity as the main flow. The large-scale mode gives the critical Reynolds number in a more isotropic case (i.e., k>0.6), while the periodic mode does so in the less isotropic case (i.e., k<0.6). Asymptotic expansions from (α,β)=(0,0) agree with the numerical results. Using the periodic mode, a possible explanation is given for the merging process of a pair of counter-rotating vortices observed in the experiments of a linear array of vortices by Tabeling et al. [J. Fluid Mech. 213, 511 (1990)]. © 1995 American Institute of Physics.