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Non-uniqueness and bifurcation in annular and planar channel flows

Published online by Cambridge University Press:  26 April 2006

M. S. Borgas  and
T. J. Pedley
M. S. Borgas
Affiliation:
CSIRO, Division of Atmospheric Research, Private Bag No. 1, Mordialloc, Victoria 3195, Australia
T. J. Pedley
Affiliation:
Department of Applied Mathematical Studies, The University, Leeds LS2 9JT, UK
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Abstract

High-Reynolds-number steady flow in an annular pipe which encounters a shallow axisymmetric expansion or indentation in the walls is studied using interactive boundary-layer theory. The flow upstream of the indentation (x < 0) is fully developed; the ratio of the shear rate on the outer wall to that on the inner wall is denoted by ρ (0 < ρ < 1): similarity solutions are found for the case where the wall perturbations are proportional to $x^{\frac{1}{3}}$. The solution is unique in a constriction, when the pressure gradient (represented by a parameter b) is favourable (b < 0). In an expansion, however, with an adverse pressure gradient, three different solutions are found if b exceeds a critical value bc. When ρ ≠ 1, one of these solutions, representing a flow that is attached on the inner wall and separated (i.e. has negative wall shear) on the outer, is a continuation of the unique doubly attached flow at small b. The other two, one separated on the inner and not the outer wall and the other separated on both walls, arise from a saddle-node bifurcation at b = bc. The doubly separated flow is never stable, as observed in diffusers. In the case of a planar channel (ρ = 1) symmetry is restored, and the non-uniqueness arises through a supercritical pitchfork bifurcation. This agrees with previous computations on channel flow, but not with Jeffery-Hamel flow, for which the bifurcation is subcritical.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 214 , May 1990 , pp. 229 - 250
Copyright
© 1990 Cambridge University Press

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References

Banks, W. H. H. & Drazin, P. G. 1973 Perturbation methods in boundary layer theory. J. Fluid Mech. 58, 763–775.Google Scholar
Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1988 On perturbation of Jeffery-Hamel flow. J. Fluid Mech. 186, 559–581.Google Scholar
Borgas, M. S. 1986 Waves, singularities and non-uniqueness in channel and pipe flows. Ph.D. dissertation, Cambridge University.
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. R. Soc. Lond. A 267, 119–138.Google Scholar
Fraenkel, L. E. 1963 Laminar flow in symmetric channels with slightly curved walls. II. An asymptotic series for the stream function.. Proc. R. Soc. Lond. A 272, 406–428.Google Scholar
Libby, P. A. & Fox, H. 1963 Some perturbation solutions in laminar boundary layer theory. Part I. The momentum equation. J. Fluid Mech. 17, 433–449.Google Scholar
Meksyn, D. 1961 New Methods in Laminar Boundary-Layer Theory. Pergamon.
Pedley, T. J. 1972 Two-dimensional boundary layers in a free stream which oscillates without reversing. J. Fluid Mech. 55, 359–384.Google Scholar
Reneau, L. R., Johnston, J. P. & Kline, S. J. 1967 Performance and design of straight, two-dimensional diffusers. Trans. ASME D: J. Basic Engng 89, 141–150.Google Scholar
Smith, F. T. 1976a Flow through constricted or dilated pipes and channels, Part I. Q. J. Mech. Appl. Maths 29, 343–364.Google Scholar
Smith, F. T. 1976b Flow through constricted or dilated pipes and channels, Part II. Q. J. Mech. Appl. Maths 29, 365–376.Google Scholar
Smith, F. T. 1977 Upstream interactions in channel flows. J. Fluid Mech. 79, 631–655.Google Scholar
Smith, F. T. 1979 The separating flow through a severely constricted symmetric tube. J. Fluid Mech. 90, 725–754.Google Scholar
Smith, F. T. 1984 Non-uniqueness in wakes and boundary layers.. Proc. R. Soc. Lond. A 391, 1–26.Google Scholar
Sobey, I. J. 1985 Observation of waves during oscillatory channel flow. J. Fluid Mech. 151, 395–426.Google Scholar
Sobey, I. J. & Drazin, P. G. 1986 Bifurcations of two-dimensional channel flows. J. Fluid Mech. 171, 263–287.Google Scholar
Stewartson, K. & Williams, P. G. 1973 On self-induced separation II. Mathematika 20, 98–108.Google Scholar
Ward-Smith, A. J. 1980 Internal Fluid Flow. Clarendon.