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Optimal energy growth and optimal control in swept Hiemenz flow

Published online by Cambridge University Press:  05 October 2006

ALAN GUÉGAN
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France
PETER J. SCHMID
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA
PATRICK HUERRE
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France

Abstract

The objective of the study is first to examine the optimal transient growth of Görtler–Hämmerlin perturbations in swept Hiemenz flow. This configuration constitutes a model of the flow in the attachment-line boundary layer at the leading-edge of swept wings. The optimal blowing and suction at the wall which minimizes the energy of the optimal perturbations is then determined. An adjoint-based optimization procedure applicable to both problems is devised, which relies on the maximization or minimization of a suitable objective functional. The variational analysis is carried out in the framework of the set of linear partial differential equations governing the chordwise and wall-normal velocity fluctuations. Energy amplifications of up to three orders of magnitude are achieved at low spanwise wavenumbers ($k\,{\sim}\,0.1$) and large sweep Reynolds number ($\hbox{\textit{Re}}\,{\sim}\,2000$). Optimal perturbations consist of spanwise travelling chordwise vortices, with a vorticity distribution which is inclined against the sweep. Transient growth arises from the tilting of the vorticity distribution by the spanwise shear via a two-dimensional Orr mechanism acting in the basic flow dividing plane. Two distinct regimes have been identified: for $k\,{\lesssim}\,0.25$, vortex dipoles are formed which induce large spanwise perturbation velocities; for $k\,{\gtrsim}\,0.25$, dipoles are not observed and only the Orr mechanism remains active. The optimal wall blowing control yields for instance an 80% decrease of the maximum perturbation kinetic energy reached by optimal disturbances at $\hbox{\textit{Re}}\,{=}\,550$ and $k\,{=}\,0.25$. The optimal wall blowing pattern consists of spanwise travelling waves which follow the naturally occurring vortices and qualitatively act in the same manner as a more simple constant gain feedback control strategy.

Type
Papers
Copyright
© 2006 Cambridge University Press

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