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On the stability of slowly varying flow: the divergent channel

Published online by Cambridge University Press:  29 March 2006

P. M. Eagles  and
M. A. Weissman
P. M. Eagles
Affiliation:
Mathematics Department, The City University, London Present address: Division of Maritime Science, The National Physical Laboratory, Teddington, Middlesex.
M. A. Weissman
Affiliation:
Mathematics Department, Imperial College, London
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Abstract

The linear stability of a slowly varying flow, the flow in a diverging straight-walled channel, is studied using a modification of the ‘WKB’ or ‘ray’ method. It is shown that ‘quasi-parallel’ theory, the usual method for handling such flows, gives the formally correct lowest-order growth rate; however, this growth rate can be substantially in error if its magnitude is comparable to that of the rate of change of the basic state. The method used clearly demonstrates the dependence of the growth rate, wavenumber, neutral curves, etc., on the cross-stream variable and on the flow quantity under consideration. When applied to the divergent channel, the method yields a much wider ‘unstable’ region and a much lower ‘critical’ Reynolds number (depending on the flow quantity used) than those predicted by quasi-parallel theory. The determination of the downstream development of waves of constant frequency shows that waves of all frequencies eventually decay.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 69 , Issue 2 , 27 May 1975 , pp. 241 - 262
Copyright
© 1975 Cambridge University Press

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References

Benney, D. J. & Rosenblat, S. 1964 Stability of spatially varying and time-dependen flows Phys. Fluids, 7, 1385.Google Scholar
Bouthier, M. 1972 Stabilité linéaire des écoulements presque parallèles J. Mécanique, 11, 99.Google Scholar
Bouthier, M. 1973 Stabilité linéaire des écoulements presque parallèles. II. La couche limite de Blasius J. Mécanique, 12, 75.Google Scholar
Bretherton, F. P. 1966 The propagation of groups of internal gravity waves in a shear flow Quart. J. Roy. Met. Soc. 92, 466.Google Scholar
Bretherton, F. P. 1968 Propagation in slowly varying waveguides. Proc. Roy. Soc. A 302, 555.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas, chap. 2. M.I.T. Press, Research Monograph, no. 29.
Drazin, P. G. 1974 On a model of instability of a slowly-varying flow Quart. J. Mech. Appl. Math. 27, 69.Google Scholar
Eagles, P. M. 1966 The stability of a family of Jeffery-Hamel solutions for divergent channel flow J. Fluid Mech. 24, 191.Google Scholar
Eagles, P. M. 1973 Supercritical flow in a divergent channel J. Fluid Mech. 57, 149.Google Scholar
Fraenkel, L. E. 1962 Laminar flow in symmetric channels with slightly curved walls. I. On the Jeffery-Hamel solutions for flow between plane walls. Proc. Roy. Soc. A 267, 119.Google Scholar
Fraenkel, L. E. 1963 Laminar flow in symmetric channels with slightly curved walls. II. An asymptotic series for the stream function. Proc. Roy. Soc. A 272, 406.Google Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer J. Fluid Mech. 22, 433.Google Scholar
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability J. Fluid Mech. 66, 465.Google Scholar
Heading, J. 1962 Introduction to Phase-Integral Methods. Methuen.
Keller, J. B. 1958 Surface waves on water of non-uniform depth J. Fluid Mech. 4, 607.Google Scholar
Lanchon, H. & Eckhaus, W. 1964 Sur l'analyse de la stabilité des écoulements faiblement divergents J. Mécanique, 3, 445.Google Scholar
Lewis, R. M. 1964 Asymptotic methods for the solution of dispersive hyperbolic equations. In Asymptotic Solutions of Differential Equations and their Applications (ed. E. Wilcox), p. 53. Wiley.
Ling, C.-H. & Reynolds, W. C. 1973 Non-parallel flow corrections for the stability of shear flows J. Fluid Mech. 59, 571.Google Scholar
Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake J. Fluid Mech. 51, 233.Google Scholar
Nayfeh, A. H., Mook, D. T. & Saric, W. S. 1974 Stability of non-parallel flows Arch. Mech. 26, 409.Google Scholar
Rosenblat, S. 1968 Centrifugal instability of time-dependent flows. Part 1. Inviscid, periodic flows J. Fluid Mech. 33, 321.Google Scholar
Rosenblat, S. & Herbert, D. M. 1970 Low-frequency modulation of thermal instability J. Fluid Mech. 43, 385.Google Scholar
Ross, J. A., Barnes, F. H., Burns, J. G. & Ross, M. A. S. 1970 The flat plate boundary layer. Part 3. Comparison of theory with experiment J. Fluid Mech. 43, 819.Google Scholar
Schlichting, H. 1968 Boundary Layer Theory, 6th edn. McGraw-Hill.
Scotti, R. S. & Corcos, G. M. 1972 An experiment on the stability of small disturbances in a stratified free shear layer J. Fluid Mech. 52, 499.Google Scholar
Shen, S. F. 1961 Some considerations on the laminar stability of time-dependent basic flows J. Aero. Sci. 28, 397.Google Scholar