Optimal energy growth and optimal control in swept Hiemenz flow
Résumé
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 ~ 2000) and large sweep Reynolds number (Re ~ 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 ? 0.25, vortex dipoles are formed which induce large spanwise perturbation velocities; for k ? 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 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. © 2006 Cambridge University Press.
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