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https://hdl.handle.net/2440/71869
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Type: | Journal article |
Title: | The Jeffery-Hamel similarity solution and its relation to flow in a diverging channel |
Author: | Haines, Philip Edward Hewitt, Richard E. Hazel, Andrew |
Citation: | Journal of Fluid Mechanics, 2011; 687:404-430 |
Publisher: | Cambridge University Press |
Issue Date: | 2011 |
ISSN: | 0022-1120 |
School/Discipline: | School of Mathematical Sciences |
Statement of Responsibility: | P.E. Haines, R.E. Hewitt and A.L. Hazel |
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 2_, 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; _/ 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 _ _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. |
Keywords: | Bifurcation; general fluid mechanics |
Rights: | © Cambridge University Press 2011 |
DOI: | 10.1017/jfm.2011.362 |
Appears in Collections: | Mathematical Sciences publications |
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