Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-19T17:46:33.359Z Has data issue: false hasContentIssue false
  • English
  • Français

Waves in parallel or swirling stratified shear flows

Published online by Cambridge University Press:  18 April 2017

S. Leibovich
S. Leibovich*
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Upson Hall, Cornell University, Ithaca, N.Y. 14853
Get access Rights & Permissions [Opens in a new window]

Extract

The dispersion relations for infinitesimal internal gravity waves (A) and axisymmetric waves in swirling streams (B) are considered. In both cases the mainstream may be sheared and density stratified in the transverse (vertical in case A, radial in case B) direction. The following results are proved for either case: If the maximum speed Wmax (or minimum speed Wmin) (in a meridian plane in case B) of the mainstream occurs at an interior point in the fluid, then the phase speed of any mode takes all values from the Wmax (or Wmin) to +∞ ( —∞) as the overall Richardson number λ2 varies from 0 to ∞. If Wmax(Wmin) is attained at a boundary point with finite rate of strain, there is a positive non-zero critical Richardson number below which one or both branches of the dispersion relation terminate. These results employ variational methods and correct erroneous results concerning problem B stated in Chandrasekhar's treatise on hydrodynamic stability. Furthermore, bounds are given on the group velocity for both branches of the dispersion relation. From these bounds it is shown that in the absence of reversals of the mainstream (Wmin > 0) upstream propagation of wave energy is impossible whenever upstream propagation of constant phase surfaces is impossible.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 93 , Issue 3 , August 1979 , pp. 401 - 412
Copyright
Copyright © Cambridge University Press 1979

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

Banks, W. H. H., Dbazin, P. G. & Zatiirska, M. B. 1976 On The Normal Modes Of Parallel Flow Of Inviscid Stratified Fluid. J. Fluid Mech. 75, 149–171.CrossRefGoogle Scholar
Bell, T. H. 1974 Effects Of Shear On The Properties Of Internal Gravity Wave Modes. Deutsche Hydrographische Zeitschrift 27, 57–62.Google Scholar
Benjamin, T. B. 1962 Theory Of The Vortex Breakdown Phenomenon. J. Fluid Mech. 14, 593–629.CrossRefGoogle Scholar
Chandbasekhab, S. 1961 Hydrodynamic And Hydromagnetic Stability. Oxford: Clarendon Press, Section 786, Pp. 366–371.Google Scholar
Coubant, R. & Hilbebt, D. 1953 Methods Of Mathematical Physics. New York: Interscience.Google Scholar
Howabd, L. N. & Gupta, A. S. 1962 On The Hydrodynamic And Hydromagnetic Stability Of Swirling Flows. J. Fluid Mech. 14, 463–476.Google Scholar
Leibovich, S. 1969 Stability Of Density Stratified Rotating Flows. A.I.A.A. J. 7, 177–178.Google Scholar
Leibovich, S. 1978 The Structure Of Vortex Breakdown. Ann. Rev. Fluid Mech. 10, 221–246.CrossRefGoogle Scholar
Maslowe, S. A. 1977 Weakly Nonlinear Stability Theory Of Stratified Shear Flows. Quart. J. Roy. Met. Soc. 103, 769–783.CrossRefGoogle Scholar
Miles, J. W. 1961 On The Stability Of Heterogeneous Shear Flows. J. Fluid Mech. 10, 496–508.Google Scholar
Squike, H. B. 1960 Analysis Of The Vortex Breakdown’ Phenomenon. Part 1. Aero. Dept., Imperial Coll., London, Rep. 102. Also In Miszellaneen Der Angewandten Mechanik, Pp. 306-12, 1962. Berlin: Akademie-Verlag.Google Scholar