On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow
- 1 November 1960
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Fluid Mechanics
- Vol. 9 (3), 353-370
- https://doi.org/10.1017/s002211206000116x
Abstract
This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number.It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.Keywords
This publication has 7 references indexed in Scilit:
- APPENDIXActa Psychiatrica Scandinavica, 1958
- On the non-linear mechanics of hydrodynamic stabilityJournal of Fluid Mechanics, 1958
- Angiomyxoma of the Left AuriclePublished by Oxford University Press (OUP) ,1957
- On the Effects of the Reynolds Stress on Hydrodynamic StabilityZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1956
- Calculated Amplified Oscillations in the Plane Poiseuille and Blasius FlowsJournal of the Aeronautical Sciences, 1954
- The Stability of Plane Poiseuille FlowPhysical Review B, 1953
- Über Stabilität und Turbulenz von FlüssigkeitsströmenAnnalen der Physik, 1924