Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-18T23:01:15.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Kelvin-Helmholtz instability of a slowly varying flow

Published online by Cambridge University Press:  29 March 2006

P. G. Drazin
P. G. Drazin
Affiliation:
Advanced Study Program, National Center for Atmospheric Research, Boulder, Colorado
Permanent address: School of Mathematics, University of Bristol, England.
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of a basic flow of two homogeneous inviscid incompressible fluids under the action of gravity is treated mathematically. In the basic state, one fluid is at rest below a horizontal plane z = 0; and the other flows above in the x direction, its speed varying slowly with the lateral co-ordinate y. The eigenvalue problem for normal modes is derived; its equation is a partial differential one, the co-ordinates y and z not being separable. The problem is solved approximately by taking the modes locally as if the basic velocity were independent of y, though the lateral wavenumber is allowed to vary slowly with y. This leads to an ordinary differential equation in y which is solved by the JWKB method. Detailed calculations are made for a parabolic profile, representing the blowing of air over water in a wide channel, and for other profiles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 65 , Issue 4 , 2 October 1974 , pp. 781 - 797
Copyright
© 1974 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

Barcilon, A. & Drazin, P. G. 1972 Geophys. Fluid Dyn. 4, 147–158.
Blumen, W. 1971 J. Atmos. Sci. 28, 340–344.
Blumen, W. 1973 Tellus, 25, 12–19.
Blumen, W. 1974 To be published.
Bouthier, M. 1972 J. Méc. 11, 599–621.
Bouthier, M. 1973 J. Méc. 12, 75–95.
Drazin, P. G. 1974 Quart. J. Mech. Appl. Math. (in Press).
Drazin, P. G. & Howard, L. N. 1966 Adv. Appl. Mech. 9, 1–89.
Fröman, N. & Fröman, P. O. 1965 JWKB Approximation. North Holland.
Gent, P. R. 1974 To be published.
Heading, J. 1962 The Phase Integral Method. Methuen.
Howard, L. N. 1961 J. Fluid Mech. 10, 509–512.
Jeffreys, H. 1962 Asymptotic Approximations. Oxford University Press.
Longuet-Higgins, M. S. & Stewart, R. W. 1961 J. Fluid Mech. 10, 529–549.