Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-20T06:55:58.610Z Has data issue: false hasContentIssue false
  • English
  • Français

On the normal modes of parallel flow of inviscid stratified fluid. Part 2. Unbounded flow with propagation at infinity

Published online by Cambridge University Press:  19 April 2006

P. G. Drazin ,
M. B. Zaturska  and
W. H. H. Banks
P. G. Drazin
Affiliation:
School of Mathematics, University of Bristol, England
M. B. Zaturska
Affiliation:
School of Mathematics, University of Bristol, England
W. H. H. Banks
Affiliation:
School of Mathematics, University of Bristol, England
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear perturbations of the flow of a non-diffusive fluid are considered. The classification of the normal modes of parallel flow of an inviscid stratified fluid presented by Banks, Drazin & Zaturska (1976) is here extended to encompass modes which propagate at infinity. When the basic flow is unbounded and the buoyancy frequency is non-zero at infinity the five classes presented earlier are augmented by three further classes: for a given flow and wavenumber they are (a) a continuous class of non-singular stable modes which are modifications of internal gravity waves by shear; (b) a continuous class of stable modes which are singular at each critical layer but otherwise similar to those of class (a); and (c) a finite number of marginally stable singular modes with over-reflexion. This classification is illustrated by many new results. Some asymptotic properties of the stable and unstable modes are found for large values of the Richardson number and for long waves. Two prototype problems, in which the basic flows are a piecewise-linear shear layer and a triangular jet, are solved analytically. The modified internal gravity waves for a Bickley jet with uniform buoyancy frequency are treated to illustrate the complementary nature of the propagating and evanescent modes. This treatment is both analytical and numerical. The general ideas are further illustrated by a numerical study of the stability characteristics of a hyperbolic-tangent shear layer. Finally the modes for a basic flow of boundary-layer type are found in exact terms of a hypergeometric function.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 95 , Issue 4 , 27 December 1979 , pp. 681 - 705
Copyright
© 1979 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

Acheson, D. J. 1976 On over-reflexion. J. Fluid Mech. 77, 433–472.Google Scholar
Alterman, Z. 1961 Kelvin-Helmholtz instability in media of variable density. Phys. Fluids 4, 1177–1179.Google Scholar
Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1976 On the normal modes of parallel flow of inviscid stratified fluid. J. Fluid Mech. 75, 149–171.Google Scholar
Bell, T. H. 1974 Effects of shear on the properties of internal gravity wave modes. Deutsche Hydrograph. Zeits. 27, 57–62.Google Scholar
Blumen, W., Drazin, P. G. & Billings, D. F. 1975 Shear layer instability of an inviscid compressible fluid. Part 2. J. Fluid Mech. 71, 305–316.Google Scholar
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513–539.Google Scholar
Brown, S. N. & Stewartson, K. 1978 The evolution of a small inviscid disturbance to a marginally unstable stratified shear flow; stage two. Proc. Roy. Soc. A 363, 175–194.Google Scholar
Drazin, P. G. 1958 The stability of a shear layer in an unbounded heterogeneous inviscid fluid. J. Fluid Mech. 4, 214–224.Google Scholar
Drazin, P. G. & Davey, A. 1977 Shear layer instability of an inviscid compressible fluid. Part 3. J. Fluid Mech. 82, 255–260.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Einaudi, F. & Lalas, D. P. 1976 The effect of boundaries on the stability of inviscid stratified shear flows. Trans. Amer. Soc. Mech. Eng. E 98, 243–248.Google Scholar
Eliassen, A., H ø ILAND, E. & Riis, E. 1953 Two-dimensional perturbation of a flow with constant shear of a stratified fluid. Inst. Weather & Climate Res., Oslo, no. 1.Google Scholar
Grimshaw, R. H. J. 1976a The reflection of internal gravity waves from a shear layer. Quart. J. Mech. Appl. Math. 29, 511–525.Google Scholar
Grimshaw, R. H. J. 1976b Nonlinear aspects of an internal gravity wave co-existing with an unstable mode associated with a Helmholtz velocity profile. J. Fluid Mech. 76, 65–83.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Howard, L. N. 1963 Neutral curves and stability boundaries in stratified flow. J. Fluid Mech. 16, 333–342.Google Scholar
Jones, W. L. 1968 Reflexion and stability of waves in stably stratified fluids with shear flow: a numerical study. J. Fluid Mech. 34, 609–624.Google Scholar
Lees, L. & Lin, C. C. 1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. N.A.C.A. Tech. Note TN 1115.Google Scholar
Leibovich, S. 1979 Waves in parallel or swirling stratified shear flows. J. Fluid Mech. 93, 401–412.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Mcintyre, M. E. & Weissman, M. A. 1978 On radiating instabilities and resonant over-reflection. J. Atmos. Sci. 35, 1190–1196.Google Scholar
Miles, J. W. 1957 On the reflection of sound at an interface of relative motion. J. Acoust. Soc. Am. 29, 226–228.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.Google Scholar
Miles, J. W. 1963 On the stability of heterogeneous shear flows. Part 2. J. Fluid Mech. 16, 209–227.Google Scholar
Miles, J. W. 1967 Internal waves in a continuously stratified atmosphere or ocean. J. Fluid Mech. 28, 305–310.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 2. McGraw-Hill.
Silcock, G. 1975 On the stability of parallel stratified shear flows. Ph.D. dissertation, University of Bristol.
Van vleck, E. B. 1902 A determination of the number of real and imaginary roots of the hypergeometric series. Trans. Am. Math. Soc. 3, 110–131.Google Scholar
Yih, C.-S. 1974 Instability of stratified flows as a result of resonance. Phys. Fluids 17, 1483–1495.Google Scholar