Large-scale instability of generalized oscillating Kolmogorov flows

The stability of an incompressible unidirectional ow that depends periodically, but otherwise arbitrarily, on a transverse coordinate and on time is considered. An iterative solution of an infinite-dimensional eigenvalue problem is constructed by a rigorous perturbation method. The critical Reynolds number R-c and the critical direction for which the large-scale "eddy viscosity" is minimum (and equal to zero) are determined by a system of two algebraic equations. For both time-independent and time-dependent cases, it turns out that the fastest-growing critical disturbances generally do not have the same transverse periodicity as that of basic ow. In the limit of large frequencies of oscillation, stability is essentially determined by the time-averaged ow. When the latter vanishes, the ow is absolutely stable for sufficiently large frequencies.