Hydrodynamic Stability Without Eigenvalues

TLDR

Fluid flows that are smooth at low speeds become unstable and turbulent at higher speeds, yet traditional eigenvalue analyses of the linearized equations often disagree with experiments, though linear effects remain central to hydrodynamic instability. The paper reconciles this discrepancy by showing that pseudospectra of the linearized equations can amplify small perturbations by roughly 10^5, even though all eigenmodes decay monotonically. The authors employ pseudospectral analysis of the linearized flow equations, revealing that nonorthogonal eigenfunctions can produce large transient growth and that the method extends to other problems with nonorthogonal eigenfunctions.

Abstract

Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem, but the results of such investigations agree poorly in many cases with experiments. Nevertheless, linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis is presented based on the "pseudospectra" of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors on the order of 10(5) by a linear mechanism even though all the eigenmodes decay monotonically. The methods suggested here apply also to other problems in the mathematical sciences that involve nonorthogonal eigenfunctions.