LOCAL ANALYSIS OF THE ONSET OF INSTABILITY IN SHEAR FLOWS

Local analysis of the onset of instability in flows that are not exactly parallel is considered. Corrections to the Orr-Sommerfeld equation arising as a consequence of the nonparallelism of the unperturbed flow are studied. The quasiparallel hypothesis is quantified on a model of Gaussian plane wave packets. It appears that the characteristic length scales of the downstream dependence of flow field characteristics must be substantially larger than the inverse of the wave number characterizing the instability. A new eigenvalue problem describing the propagation of these Gaussian wave packets is written. The relation between the marginal and absolute instability analysis for marginal Reynolds numbers is discussed. For flows varying slowly in the downstream direction, closed-form corrections of the Orr-Sommerfeld equation terms taking account of the x variation of the flow field and of the extension of the propagating wave packets are derived. A first-order perturbation theory correction of the Orr-Sommerfeld dispersion relation is proposed, allowing the reduction of the calculation of nonparallel corrections of the local instability quantities to quadratures. The proposed theory is applied to two important cases: the Blasius boundary layer and the cylinder wake. For the Blasius boundary layer the basic condition of applicability of the quasiparallel theory is found to be satisfied. However, the nonparallel correction of the critical Reynolds number is found to be non-negligible and provides a good agreement with experimental results. In the cylinder wake case direct bidimensional simulation results are used to assess the downstream variation of the flow field characteristics. The characteristic length scale of this variation in the near wake is found to be of the order of unity, which is also the magnitude of the wave numbers characterizing the local absolute instabilities in this region. Hence, the Orr-Sommerfeld analysis and any corrections based on the propagation of plane waves in the wake can hardly be expected to provide more than qualitative results.