Optimal control theory is used to determine the wall transpiration (unsteady blowing/suction) with zero net mass flux capable of attenuating Tollmien–Schlichting waves in a spatially developing boundary layer. The flow state is determined from the parabolized stability equations, in a linear setting. An appropriate cost functional is introduced and minimized iteratively by the numerical solution of the equations for the state and the dual state, coupled via transfer and optimality conditions. Central to the control is the determination of the wall Green’s function expressing the receptivity of the flow to wall inhomogeneities. The optimal wall velocity is obtained in few iterations and a reduction of several orders of magnitude in output disturbance energy is demonstrated, as compared to the uncontrolled case, for control laws operating both over the whole wall length and over a finite strip. Finally, white noise disturbances are applied to the optimal wall velocities already determined, to assess the influence of an imperfectly operating controller on the final result, and to decide on the practical feasibility of the approach.

A methodology for determining the optimal steady suction distribution for the delay of transition in a boundary layer is presented. The flow state is obtained from the coupled system of boundary layer equations and parabolized stability equations ͑PSE͒, to account for the spatially developing nature of the flow. The wall suction is defined by an optimal control procedure based on the iterative solution of the equations for the state and the dual state; the latter is available from the adjoint boundary layer equations and the adjoint PSE. The technique is applied to the control of two-dimensional Tollmien-Schlichting ͑TS͒ waves. Results show that the onset of the instability can be significantly postponed and/or the growth rate considerably reduced by applying an appropriate suction through the whole wall length, in a wide frequency band. Control over panels of finite length completes the study and brings useful, preliminary information on the practicality of the approach in view of implementation. Finally, a simplified methodology which does not rely on the PSE is discussed, based on the minimization of the shape factor. Satisfactory results are achieved with this simpler approach which might, thus, constitute a method of choice when results are needed rapidly, i.e., during on-line control of TS waves.

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The relation between the receptivity and the sensitivity of the incompressible flow in the boundary layer over a flat plate to harmonic perturbations is determined. Receptivity describes the birth of a disturbance, whereas sensitivity is a concept of larger breath, describing the modification incurred by the state of a system as a response to parametric variations. The governing equations ruling the system's state are the non-local stability equations. Receptivity and sensitivity functions can be obtained from the solution of the adjoint system of equations. An application to the case of Tollmien-Schlichting waves spatially developing in a flat plate boundary layer is studied. To cite this article: C. Airiau et al., C. R. Mecanique 330 (2002) 259-265.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS fluid mechanics / stability / adjoint / receptivity / sensitivity Sensibilité et réceptivité d'une couche limite Résumé La relation entre les fonctions de réceptivité et de sensibilité d'une couche limite incompressible à des perturbations du type sources harmoniques est démontrée. La réceptivité décrit la naissance d'une onde d'instabilité alors que la sensibilité représente la modification de l'état d'un système à une variation d'un de ses paramètres. L'évolution spatiale des ondes d'instabilité (l'état du système) est donnée par la solution des équations de stabilité non locales. Les fonctions de réceptivité et de sensibilité sont déduites de la solution des équations adjointes. La théorie est appliquée aux ondes de Tollmien-Schlichting qui se développent spatiallement dans la couche limite de Blasius. Pour citer cet article : C. Airiau et al., C. R. Mecanique 330 (2002) 259-265.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS mécanique des fluides / stabilité / adjoint / réceptivitité / sensibilité

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