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Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

Published online by Cambridge University Press:  16 October 2008

UWE EHRENSTEIN  and
FRANÇOIS GALLAIRE
UWE EHRENSTEIN
Affiliation:
IRPHÉ UMR 6594, Aix-Marseille Université, CNRS, 49 Rue Joliot-Curie, F-13384 Marseille Cedex 13, [email protected]
FRANÇOIS GALLAIRE
Affiliation:
Laboratoire J. A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 02, [email protected]
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Abstract

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 614 , 10 November 2008 , pp. 315 - 327
Copyright
Copyright © Cambridge University Press 2008

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