SIDE-WALL EFFECT ON THE LONG-WAVE INSTABILITY IN KOLMOGOROV FLOW

Linear stability of the Kolomogrov flow U(y), which shows the long-wav e instability in unbounded region, is investigated in restricted regio n {-infinity < x < infinity, 0 less than or equal to y less than or eq ual to L pi} under two kinds of boundary conditions: a real viscous bo undary condition and a virtual stress-free boundary condition. It is s hown numerically and analytically that a finite-wave instability appea rs instead of the long-wave instability owing to the confinement. The numerical results suggest that the critical Reynolds number R-c(L) and the critical wavenumber alpha(c)(L) tend to those in unbounded region as R-c(L) - R-c(infinity) proportional to L-1, alpha(c)(L) - alpha(c) (infinity)(= 0) proportional to L-0.5. This power law is different fro m that for the finite-wavenumber instability. The effect of the differ ence of the boundary condition on the stability is also elucidated. It is shown numerically that it does not affect the critical values exce pt for small systems (L < 5).