The Jeffery-Hamel similarity solution and its relation to flow in a diverging channel

Haines, P.E. and Hewitt, R.E. and Hazel, A.L. (2011) The Jeffery-Hamel similarity solution and its relation to flow in a diverging channel. Journal of Fluid Mechanics, 687. pp. 404-430. ISSN 1469-7645

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Abstract

We explore the relevance of the idealized Jeffery�Hamel similarity solution to the practical problem of flow in a diverging channel of finite (but large) streamwise extent. Numerical results are presented for the two-dimensional flow in a wedge of separation angle 2α, bounded by circular arcs at the inlet/outlet and for a net radial outflow of fluid. In particular, we show that in a finite domain there is a sequence of nested neutral curves in the (Re, α) plane, each corresponding to a midplane symmetry-breaking (pitchfork) bifurcation, where Re is a Reynolds number based on the radial mass flux. For small wedge angles we demonstrate that the first pitchfork bifurcation in the finite domain occurs at a critical Reynolds number that is in agreement with the only pitchfork bifurcation in the infinite-domain similarity solution, but that the criticality of the bifurcation differs (in general). We explain this apparent contradiction by demonstrating that, for α 1, superposition of two (infinite-domain) eigenmodes can be used to construct a leading-order finite-domain eigenmode. These constructed modes accurately predict the multiple symmetry-breaking bifurcations of the finite-domain flow without recourse to computation of the full field equations. Our computational results also indicate that temporally stable, isolated, steady solutions may exist. These states are finite-domain analogues of the steady waves recently presented by Kerswell, Tutty, & Drazin (J. Fluid Mech., vol. 501, 2004, pp. 231�250) for an infinite domain. Moreover, we demonstrate that there is non-uniqueness of stable solutions in certain parameter regimes. Our numerical results tie together, in a consistent framework, the disparate results in the existing literature.

Item Type:	Article
Additional Information:	© 2011 Cambridge University Press
Uncontrolled Keywords:	Expanding Channel Flow Jeffery-Hamel Bifurcation Symmetry Breaking
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 76 Fluid mechanics PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 40 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS > 47 Fluid dynamics
Depositing User:	Dr Richard E. Hewitt
Date Deposited:	14 Nov 2013
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2045

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