Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-18T20:58:46.820Z Has data issue: false hasContentIssue false
  • English
  • Français

The instability to long waves of unbounded parallel inviscid flow

Published online by Cambridge University Press:  28 March 2006

P. G. Drazin  and
L. N. Howard
P. G. Drazin
Affiliation:
Mathematics Department, University of Bristol
L. N. Howard
Affiliation:
Mathematics Department, Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

Formulas for the determination of the instability characteristics of unbounded parallel flow are obtained for the case of long waves, and applied, together with some general results, to give a qualitative description of the different modes of instability of such flows. It is found that there is a finite number of different modes unstable to long waves, essentially one for each relative maximum and minimum of the velocity profile. These modes appear to become stable when the wavelength is sufficiently small, reducing to neutral solutions associated with inflexion points as stability is approached. The formulas are also useful for quantitative calculation of instability characteristics.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 14 , Issue 2 , October 1962 , pp. 257 - 283
Copyright
© 1962 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

Carrier, G. F. & Chang, C. T. 1959 Quart. Appl. Math. 16, 436–9.
Case, K. M. 1960a Phys. Fluids 3, 143–8.
Case, K. M. 1960b Phys. Fluids 3, 149–54.
Case, K. M. 1960c Phys. Fluids 3, 432–5.
Case, K. M. 1961 J. Fluid Mech. 10, 420–9.
Coddington, E. A. & Levinson, N. 1955 Theory of Ordinary Differential Equations. New York: McGraw-Hill.
Curle, N. 1956a Aero. Res. Council, Lond., Unpublished Rep. no. 18426.
Curle, N. 1956b Aero. Res. Council, Lond., Unpublished Rep. no. 18564.
Eliassen, A., Höiland, E. & Riis, E. 1953 Institute Weather and Climate Research, Oslo, Publ. no. 1.
Esch, R. E. 1957 J. Fluid Mech. 3, 289–303.
Friedrichs, K. O. 1942 Fluid Dynamics (mimeographed lecture notes, Brown University), Chap. VI.
Garcia, R. V. 1956 Tellus, 8, 82–93.
Hollingdale, S. H. 1940 Phil. Mag. (7), 29, 209–57.
Howard, L. N. 1959 J. Math. Phys. 37, 283–98.
Howard, L. N. 1961 J. Fluid Mech. 10, 509–12.
Lessen, M. & Fox, J. A. 1955 50 Jahre Grenzschicht-Forschung, pp. 122–126. Braunschweig: Friedr. Vieweg und Sohn.
Lighthill, M. J. 1957 J. Fluid Mech. 3, 113–44.
Lin, C. C. 1953 Nat. Adv. Comm. Aero., Wash., Tech. Note no. 2887.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Miles, J. W. 1957 J. Fluid Mech. 3, 185–204.
Orr, W. M'F. 1907 Proc. Roy. Irish Acad. A, 28, 9–68.
Rayleigh, J. W. S. 1945 The Theory of Sound, vol. II, chap. XXI (reprint of 2nd ed. of 1894). New York: Dover.
Sato, H. 1960 J. Fluid Mech. 7, 53–80.
Savic, P. 1941 Phil. Mag. (7), 32, 245–52.
Savic, P. & Murphy, J. W. 1943 Phil. Mag. (7), 34, 139–44.
Squire, H. B. 1933 Proc. Roy. Soc. A, 142, 621–8.
Synge, J. L. 1933 Trans. Roy. Soc. Canada, 27 (III), 1–18.
Tatsumi, T. & Gotoh, K. 1960 J. Fluid Mech. 7, 433–41.
Tatsumi, T. & Kakutani, T. 1958 J. Fluid Mech. 4, 261–75.
Tollmien, W. 1935 Nach. Ges. Wiss. Göttingen, Math.-phys. Klasse, 50, 79–114.