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The stability of a family of Jeffery–Hamel solutions for divergent channel flow

Published online by Cambridge University Press:  28 March 2006

P. M. Eagles
P. M. Eagles
Affiliation:
Northampton College of Advanced Technology, London, E.C. 1
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Abstract

A set of Jeffery–Hamel profiles (for radial, viscous, incompressible flow) have been shown by Fraenkel (1962, 1963) to approximate to profiles in certain two-dimensional divergent channels. The stability of a family of these profiles is investigated by a numerical solution of the Orr-Sommerfeld problem. Neutralstability curves are calculated in the (R,k)-planes (where R is the Reynolds number of the basic flow and k is the wave-number of the disturbance), and fairly low critical Reynolds numbers are found. For those profiles that have regions of reversed flow, negative wave velocities are found on the lower branch of the neutral curve, and also it is found that Rk tends to a finite limit as R → ∞ on the lower branch. These unexpected results are further discussed and verified by independent methods. The relation of the calculations to some experiments of Patterson (1934, 1935) is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 24 , Issue 1 , January 1966 , pp. 191 - 207
Copyright
© 1966 Cambridge University Press

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References

Fraenkel, L. E. 1962 Proc. Roy. Soc., A 267, 119.
Fraenkel, L. E. 1963 Proc. Roy. Soc., A 272, 406.
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Kurtz, E. F. & Crandall, S. H. 1962 J. Math. Phys. 41, 26.
Lin, C. C. 1945 Quart. Appl. Math. 3, Pts. I, II, III, pp. 117, 218, 277.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Patterson, G. N. 1934 Canad. J. Res. 11, 77.
Patterson, G. N. 1935 Canad. J. Res. 12, 67.
Thomas, L. H. 1953 Phys. Rev. (2) 91, 780.