The effect of base flow variation on flow stability

TLDR

The Orr–Sommerfeld operator’s eigenvalues govern the stability of disturbances in parallel and quasi‑parallel flows, and base‑flow variations can represent differences between laboratory and ideal theoretical flows. The study evaluates how non‑infinitesimal base‑flow modifications affect eigenvalue sensitivity and motivates a velocity‑based definition of pseudospectra in hydrodynamic stability. The most destabilizing base‑flow change is identified via variational techniques applied to plane Couette flow. Small base‑flow changes destabilize the flow, despite the ideal flow being linearly stable, and these results motivate a velocity‑based pseudospectra definition. The worst case scenario is identified.

Abstract

The Orr–Sommerfeld operator's eigenvalues determine the stability of exponentially growing disturbances in parallel and quasi-parallel flows. This work assesses the sensitivity of these eigenvalues to modifications of the base flow, which need not be infinitesimally small. Such base flow variations may represent differences between the laboratory flow and its ideal, theoretical counterpart. The worst case, i.e. the change in base flow with the most destabilizing effect on the eigenvalues, is found using variational techniques for the plane Couette flow. Relatively small changes in the base flow are shown to be destabilizing, although the ideal flow is unconditionally stable according to linear theory. These observations inspire a velocity-based definition of pseudospectra in the hydrodynamic stability context.