Hostname: page-component-6587cd75c8-4pd2k Total loading time: 0 Render date: 2025-04-24T05:25:21.197Z Has data issue: false hasContentIssue false
  • English
  • Français

The spatial viscous instability of axisymmetric jets

Published online by Cambridge University Press:  11 April 2006

Philip J. Morris
Philip J. Morris
Affiliation:
Lockheed-Georgia Company, Marietta, Georgia
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of three axisymmetric jet profiles is reviewed. These profiles represent the development of an incompressible jet from a nearly top-hat profile to a fully developed jet profile. The disturbance equations for arbitrary mode number in a region of zero shear, which provide the boundary conditions for the numerical solution, are solved analytically through use of the disturbance vorticity equations. Numerical solutions for the spatial stability for the axisymmetric (n = 0) disturbance and the asymmetric n = l disturbance are presented. Previously published calculations of least stable modes are shown to be incorrectly interpreted and their actual mode types are given. The critical Reynolds number is found to increase as the profile varies from a top-hat to a fully developed jet form. Closed contours of constant amplification, which are unusual in free shear flows, are shown to exist for the n = 1 disturbance in the fully developed jet region. A fluctuation energy balance is used to justify the occurrence of this destabilizing effect of decreasing Reynolds number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 77 , Issue 3 , 8 October 1976 , pp. 511 - 529
Copyright
© 1976 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.)

Article purchase

Temporarily unavailable

References

Andbade, E. N. D. & Tsien, L. C. 1937 The velocity distribution in a liquid-into-liquid jet. Proc. Phys. Soc. Lond. 49, 381.Google Scholar
Batchelob, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529.Google Scholar
Bellman, R. E. & Kalaba, R. E. 1965 Quosi-linearizotion and Non-linear Boundary-value Problems. Elsevier.
Bubbidgb, D. M. 1968 The stability of round jets. Ph.D. thesis, Bristol University.
Conte, S. D. 1966 The numerical solution of linear boundary-value problems. Siam Rev. 8, 309.Google Scholar
Ceighton, D. G. 1973 Instability of an elliptic jet. J. Fluid Mech. 59, 665.Google Scholar
Crow, S. C. & Champagne, F. M. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48, 547.Google Scholar
Davey, A. 1973 A simple numerical method for solving Orr-Sommerfeld problems. Quart. J. Mech. Appl. Math. 26, 401.Google Scholar
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701.Google Scholar
Fbeymuth, P. 1966 On transition in a separated laminar boundary layer. J. Fluid Mech. 25, 683.Google Scholar
Fyfe, D. J. 1966 Economical evaluation of Runge-Kutta formulae. Math. Comp. 20, 392.Google Scholar
Gabg, V. K. 1971 Stability of pipe Poiseuille flow to infinitesimal disturbances. Ph.D. thesis, Carnegie-Mellon University Pittsburgh.
Gabg, V. K. & Rouleau, W. T. 1972 Linear spatial stability of pipe Poiseuille flow. J. Fluid Mech. 54, 113.Google Scholar
Gasteb, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222.Google Scholar
Gill, A. E. 1962 On the occurrence of condensations in steady axisymmetric jets. J. Fluid Mech. 14, 557.Google Scholar
Gill, A. E. 1965 On the behaviour of small disturbances to Poiseuille flow in a circular pipe. J. Fluid Mech. 21, 145.Google Scholar
Kambe, T. 1969 The stability of an axisymmetric jet with a parabolic profile. J. Phys. Soc. Japan, 26, 566.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.
Lessen, M., Sadleb, S. G. & Liu, T.-Y. 1968 Stability of pipe Poiseuille flow, Phys. Fluids, 11, 1404.Google Scholar
Lessen, M. & Singh, P. J. 1973 The stability of axisymmetric free shear layers. J. Fluid Mech. 60, 443.Google Scholar
Mattingly, G. E. & Chang, C. C. 1974 Unstable waves on an axisymmetric jet column. J. Fluid Mech. 65, 541.Google Scholar
Michalke, A. 1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, 521.Google Scholar
Michalke, A. 1971 Instabilität eines kompressiblen runden Freistrahls unter Berück-sichtigung des Einflusses der Strahlgrenzschichtdicke. Z. Flugwiss. 19, 319.Google Scholar
Mollendobf, J. C. & Gebhart, B. 1973 An experimental and numerical study of the viscous stability of a round laminar vertical jet with and without thermal buoyancy for symmetric and asymmetric disturbances. J. Fluid Mech. 61, 367.Google Scholar
Morris, P. J. 1971 The structure of turbulent shear flow. Ph.D. thesis, Southampton University.
Obszag, S. A. & Cbow, S. C. 1970 Instability of a vortex sheet leaving a semi-infinite plate. Stud. Appl. Math. 49. 167.Google Scholar
Pai, S. I. & Hsieh, T. 1972 Numerical solution of laminar jet mixing with and without free stream. Appl. Sci. Res. 27, 39.Google Scholar
Reynolds, A. J. 1962 Observations of a liquid-into-liquid jet. J. Fluid Mech. 14, 552.Google Scholar
Sharma, R. 1968 The structure of turbulent shear flow. Ph.D. thesis, Southampton University.
Viilu, A. 1962 An experimental determination of the minimum Reynolds number for instability in a free jet. J. Appl. Mech. 29, 506.Google Scholar