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Spatio-temporal structure of the ‘fully developed’ transitional flow in a symmetric wavy channel. Linear and weakly nonlinear stability analysis

Published online by Cambridge University Press:  29 December 2014

S. Blancher ,
Y. Le Guer  and
K. El Omari
S. Blancher*
Affiliation:
Université de Pau et des Pays de l’Adour, Laboratoire SIAME, Fédération CNRS IPRA, Avenue de l’Université, 64000 Pau, France
Y. Le Guer
Affiliation:
Université de Pau et des Pays de l’Adour, Laboratoire SIAME, Fédération CNRS IPRA, Avenue de l’Université, 64000 Pau, France
K. El Omari
Affiliation:
Université de Pau et des Pays de l’Adour, Laboratoire SIAME, Fédération CNRS IPRA, Avenue de l’Université, 64000 Pau, France
*
†Email address for correspondence: [email protected]
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Abstract

This work addresses the transition from 2D steady to 2D unsteady laminar flow for a fully developed regime in a symmetric wavy channel geometry. We investigate the existence and characteristics of the spatio-temporal structure of the fully developed unsteady laminar flow for those particular geometries for which the steady flow presents a periodic variation of the main stream velocity component. We perform a 2D global linear stability analysis of the fully developed steady laminar flow, and we show that, for all the geometries studied, the transition is triggered by a Hopf bifurcation associated with the breaking of the symmetries and the invariance of the steady flow. Critical Reynolds numbers, most unstable modes and their characteristics are presented for large ranges of the geometric parameters, namely wavenumber ${\it\alpha}$ from 0.3 to 5 and amplitude from 0 (straight channel) to 0.5. We show that it is possible to define geometries for which the wavenumber is proportional to the most unstable mode wavenumber for the critical Reynolds number. From this modal study we address a weakly nonlinear stability analysis with a view to obtaining the Landau coefficient $g$, and then the sub- or supercritical nature of the first bifurcation characterising the transition. We show that a critical geometric amplitude beyond which the first bifurcation is supercritical is associated with each geometric wavenumber.

JFM classification

Instability: Instability Instability: Parametric instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 764 , 10 February 2015 , pp. 250 - 276
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Copyright
© 2014 Cambridge University Press 

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