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Bifurcations and convective instabilities of steady flows in a constricted channel

Published online by Cambridge University Press:  26 June 2018

João A. Isler Open the ORCID record for João A. Isler [Opens in a new window] ,
Rafael S. Gioria  and
Bruno S. Carmo
João A. Isler*
Affiliation:
Department of Mechanical Engineering, Escola Politécnica, University of São Paulo, Av. Prof. Mello Moraes, 2231, 05508-030, São Paulo, Brazil
Rafael S. Gioria
Affiliation:
Department of Mining and Petroleum Engineering, Escola Politécnica, University of São Paulo, Av. Prof. Mello Moraes, 2373, 05508-030, São Paulo, Brazil
Bruno S. Carmo
Affiliation:
Department of Mechanical Engineering, Escola Politécnica, University of São Paulo, Av. Prof. Mello Moraes, 2231, 05508-030, São Paulo, Brazil
*
†Email address for correspondence: [email protected]
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Abstract

Stability and nonlinear analyses were employed to study symmetric and asymmetric steady flows through a straight channel with a smooth constriction with 50 % occlusion. Linear stability analysis was carried out to determine the unstable regions and the critical Reynolds numbers for the primary and secondary global instabilities. The primary bifurcation demonstrated an intricate aspect: the three-dimensional modes transfer their energy to the two-dimensional mode, which causes a symmetry breaking of the flow. This behaviour could be observed for Reynolds number lower than the critical, showing that this primary bifurcation is hysteretical. The secondary bifurcation also presented subcritical behaviour, exhibiting a pitchfork diagram with a large hysteretic curve. Given the subcritical character of both bifurcations, the relevance of non-normal growth in these flows were assessed, so that their convective mechanisms were exhaustively investigated. In addition, we could show that, for the secondary instability, optimal initial disturbances with large enough initial energy were able to promote a subcritical nonlinear saturation for a Reynolds number lower than the critical. The physical mechanism behind the transition process occurred by nonlinear interaction between the two- and three-dimensional modes, which established oscillatory behaviour, moreover, this energy exchange between the modes led the flow to the nonlinear saturated state. Therefore, the two-dimensional modes play a key role in the primary and secondary bifurcations of this system.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Waves/Free-surface Flows: Channel flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 777 - 804
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Copyright
© 2018 Cambridge University Press 

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