Optimal perturbations and minimal defects

Abstract

Two possible initial paths of transition to turbulence in simple shear flows are examined. The first is the – by now classical – transient (or algebraic) growth scenario which may have an important role in the by-pass transition of those flows for which traditional eigen-analysis predicts asymptotic stability. Transient growth is optimally excited by certain initial disturbances now known as "optimal perturbations"; they can be found through a classical variational analysis initiated by Farrell (1988). The second path starts with the exponential amplification, in nominally subcritical conditions, of modal disturbances developing over a base flow mildly distorted with respect to its idealized counterpart. The base flow distortion of given norm that excites the largest growth of the instability wave is called the “minimal defect", and its study was initiated by Bottaro et al. (2003). Both paths provide feasible initial conditions for the transition process and it is likely that in most practical situations algebraic and exponential growth mechanisms are concurrently at play in provoking transition to turbulence in shear flows.