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.
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July 2001
Research Article|
July 01 2001
Optimal control of Tollmien–Schlichting waves in a developing boundary layer
Steeve Walther;
Steeve Walther
Institut de Mécanique des Fluides de Toulouse, Allée du Professeur Camille Soula, 31400 Toulouse, France
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Christophe Airiau;
Christophe Airiau
Institut de Mécanique des Fluides de Toulouse, Allée du Professeur Camille Soula, 31400 Toulouse, France
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Alessandro Bottaro
Alessandro Bottaro
Institut de Mécanique des Fluides de Toulouse, Allée du Professeur Camille Soula, 31400 Toulouse, France
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Physics of Fluids 13, 2087–2096 (2001)
Article history
Received:
June 28 2000
Accepted:
March 19 2001
Citation
Steeve Walther, Christophe Airiau, Alessandro Bottaro; Optimal control of Tollmien–Schlichting waves in a developing boundary layer. Physics of Fluids 1 July 2001; 13 (7): 2087–2096. https://doi.org/10.1063/1.1378035
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