Hydrodynamic Stability Without Eigenvalues

• Published in: Science (American Association for the Advancement of Science) Vol. 261; no. 5121; pp. 578 - 584
• Main Authors: Trefethen, Lloyd N., Trefethen, Anne E., Reddy, Satish C., Driscoll, Tobin A.
• Format: Journal Article
• Language: English
• Published: Washington, DC American Society for the Advancement of Science 30-07-1993; American Association for the Advancement of Science; The American Association for the Advancement of Science
• Subjects: Algebra; Couette flows; Eigenvalues; Exact sciences and technology; Flow stability; Fluid dynamics; Fundamental areas of phenomenology (including applications); Hydrodynamic stability; Hydrodynamics; Hydrofoil boats; Laboratory Experiments; Laminar flow; Linear transformations; Mathematical functions; Mathematical vectors; Mathematics Instruction; Numbers; Personality; Physics; Reynolds number; Shear flow; Statistical Data; Turbulence; Poiseuille flow; Shear flow; Hydrodynamic instability; Couette flow; Perturbation; Modeling; Turbulent laminar transition
• ISSN: 0036-8075; 1095-9203
• DOI: 10.1126/science.261.5121.578

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.