Flow Instability: Transition and Non Linearity

We utilize input-output analysis and the small gain theorem to derive a disturbance-based criterion for analyzing the stability of shear flows and boundary layers. This criterion provides a bound on the magnitude of velocity perturbations, ensuring the stability of the flow. Three canonical base flows of high importance in transition analysis are used to test our criterion: Couette flow, plane Poiseuille flow, and Blasius boundary layer. We demonstrate that our analysis converges to linear stability theory when an infinitesimally small threshold is imposed for the cases of plane Poiseuille and Blasius flows. For the case of Couette flow, we utilize our stability criterion to demonstrate that this base flow can become unstable, and transition can be triggered at different Reynolds numbers, which is consistent with experimental observations. Our stability analysis allows us to account for finite-amplitude perturbations and nonlinear interactions, which enables us to explain the observed transition in various shear flows at both sub-critical and post-critical Reynolds numbers, as observed in experimental and direct numerical simulation studies.