Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-06T08:06:24.950Z Has data issue: false hasContentIssue false
  • English
  • Français

On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow

Published online by Cambridge University Press:  28 March 2006

J. T. Stuart
J. T. Stuart
Affiliation:
National Physical Laboratory, Teddington, Middlesex
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 9 , Issue 3 , November 1960 , pp. 353 - 370
Copyright
© 1960 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gorkov, L. P. 1957 J. Exp. Theor. Phys., Moscow, 33, 402; translated as Sov. Phys. J.E.T.P. 6 (33), 311 (1958).
Heisenberg, W. 1924 Ann. Phys., Lpz., (4), 74, 577; NACA TM 1291.
Ince, E. L. 1956 Ordinary Differential Equations. New York: Dover.
Landau, L. 1944 C.R. Acad. Sci. U.R.S.S. 44, 311.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lin, C. C. 1958 Boundary Layer Research, p. 144. I.U.T.A.M. Symposium, Freiburg, 1957 (editor, H. Görtler). Berlin: Springer.
Malkus, W. V. R. & Veronis, G. 1958 J. Fluid Mech. 4, 225.
Meksyn, D. & Stuart, J. T. 1951 Proc. Roy. Soc. A, 208, 517.
Noether, F. 1921 Z. angew. Math. Mech. 1, 125.
Shen, S. F. 1954 J. Aero. Sci. 21, 62–4.
Stuart, J. T. 1956a J. Aero. Sci. 23, 86, 894.
Stuart, J. T. 1956b Z. angew. Math. Mech., Sonderheft (1956), p. S. 32.
Stuart, J. T. 1958 J. Fluid Mech. 4, 1.
Stuart, J. T. 1960 On three-dimensional non-linear effects in the stability of parallel flows. Preprint for Int. Congr. Aero. Sciences, Zürich, 1960.
Stuart, J. T. & Watson, J. 1960 On the growth of finite-amplitude thermal convection. (To be published.)
Thomas, L. H. 1953 Phys. Rev. (2), 91, 780.
Veronis, G. 1959 J. Fluid Mech. 5, 401.
Watson, J. 1960 J. Fluid Mech. 9, 371.