Transient growth and minimal defects: Two possible initial paths of transition to turbulence in plane shear flows

Abstract

Two possible initial paths of transition to turbulence in simple wall-bounded shear flows are examined by looking at the development in space of infinitesimal disturbances. The first is the—by-now-classical—transient growth scenario which may have an important role in the bypass transition of flows for which traditional eigenmode analysis predicts asymptotic stability. It is studied by means of a simplified parabolic model justified by the underlying physics of the problem; results for optimal disturbances and maximum transient growth are found in excellent agreement with computations based on the full Orr–Sommerfeld/Squire equations. The second path starts with the exponential amplification, in nominally subcritical conditions, of modal disturbances superposed to base flows mildly distorted compared to their idealized counterparts. Such mean flow distortions might arise from the presence of unwanted external forcing related, for example, to the experimental environment. A technique is described that is capable of providing the worst case distortion of fixed norm for any ideal base flow, i.e., that base flow modification capable of maximizing the amplification rate of a given instability mode. Both initial paths considered here 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 destabilizing plane shear flows.