Hydrodynamic stability without eigenvalues
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 100,000 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.
 Publication:

Science
 Pub Date:
 July 1993
 DOI:
 10.1126/science.261.5121.578
 Bibcode:
 1993Sci...261..578T
 Keywords:

 Computational Fluid Dynamics;
 Eigenvalues;
 Flow Stability;
 Hydrodynamic Equations;
 Flow Equations;
 Flow Geometry;
 Laminar Flow;
 Spectral Methods;
 Transition Flow;
 Velocity Distribution;
 Vortices;
 Fluid Mechanics and Heat Transfer