Optimal energy growth and optimal control in swept Hiemenz flow

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.