EFFECT OF LATERAL BOUNDARIES ON LARGE-SCALE MODE - LINEAR-STABILITY OF SQUARE CELL FLOWS IN RECTANGULAR REGIONS

Linear stability of the square cell flow represented by the stream fun ction: psi = sin x sin y is investigated numerically in various bounde d region D = [0; M pi] x [0, N pi]. The disturbances are limited to tw o-dimensional ones and a perfect slip condition is assumed to be appli ed. Special attention is paid to clarify how the critical long-wave mo de (or large-scale mode) of the flow in unbounded region is modified b y lateral boundaries. It is shown that the critical modes are classifi ed into three cases according to the configuration (M,N): (i) M = 1, ( ii) (M,N) = (2, odd numbers), (3,4) and (3; 5), and (iii) the others. The last one is the most typical cases and is related to the long-wave mode in the unbounded region. The structure of the mode is one statio nary vortex with the system size, which we call global rotation, for M similar to N while it is a series of stationary counter-rotating vort ices for M << N. In case (ii) the critical modes are oscillatory thoug h they are related to case (iii). In case (i) (linear array of vortice s) the mode also shows the global rotation, but it is not related to t he long-wave mode in the unbounded region.