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Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities

Published online by Cambridge University Press:  10 March 2009

OLIVIER MARQUET ,
MATTEO LOMBARDI ,
JEAN-MARC CHOMAZ ,
DENIS SIPP  and
LAURENT JACQUIN
OLIVIER MARQUET
Affiliation:
Departement d'Aerodynamique Fondamentale et Experimentale, ONERA, 92190 Meudon, France
MATTEO LOMBARDI
Affiliation:
Departement d'Aerodynamique Fondamentale et Experimentale, ONERA, 92190 Meudon, France
JEAN-MARC CHOMAZ
Affiliation:
Departement d'Aerodynamique Fondamentale et Experimentale, ONERA, 92190 Meudon, France Laboratoire d'Hydrodynamique (LadHyx), CNRS - Ecole Polytechnique, 91128 Palaiseau, France
DENIS SIPP
Affiliation:
Departement d'Aerodynamique Fondamentale et Experimentale, ONERA, 92190 Meudon, France
LAURENT JACQUIN
Affiliation:
Departement d'Aerodynamique Fondamentale et Experimentale, ONERA, 92190 Meudon, France
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Abstract

The stability of the recirculation bubble behind a smoothed backward-facing step is numerically computed. Destabilization occurs first through a stationary three-dimensional mode. Analysis of the direct global mode shows that the instability corresponds to a deformation of the recirculation bubble in which streamwise vortices induce low- and high-speed streaks as in the classical lift-up mechanism. Formulation of the adjoint problem and computation of the adjoint global mode show that both the lift-up mechanism associated with the transport of the base flow by the perturbation and the convective non-normality associated with the transport of the perturbation by the base flow explain the properties of the flow. The lift-up non-normality differentiates the direct and adjoint modes by their component: the direct is dominated by the streamwise component and the adjoint by the cross-stream component. The convective non-normality results in a different localization of the direct and adjoint global modes, respectively downstream and upstream. The implications of these properties for the control problem are considered. Passive control, to be most efficient, should modify the flow inside the recirculation bubble where direct and adjoint global modes overlap, whereas active control, by for example blowing and suction at the wall, should be placed just upstream of the separation point where the pressure of the adjoint global mode is maximum.

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Papers
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Journal of Fluid Mechanics , Volume 622 , 10 March 2009 , pp. 1 - 21
Copyright
Copyright © Cambridge University Press 2009

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