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  • 1.
    Bagheri, Shervin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Brandt, Luca
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Matrix-free methods for the stability and control of boundary layers2009Ingår i: AIAA Journal, ISSN 0001-1452, E-ISSN 1533-385X, Vol. 47, nr 5, s. 1057-1068Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper presents matrix-free methods for the stability analysis and control design of high-dimensional systems arising from the discretized linearized Navier-Stokes equations. The methods are applied to the two-dimensional spatially developing Blasius boundary-layer. A critical step in the process of systematically investigating stability properties and designing feedback controllers is solving very large eigenvalue problems by storing only velocity fields at different times instead of large matrices. For stability analysis, where the entire dynamics of perturbations in space and time is of interest, iterative and adjoint-based optimization techniques are employed to compute the global eigenmodes and the optimal initial conditions. The latter are the initial conditions yielding the largest possible energy growth over a finite time interval. The leading global eigenmodes take the shape of Tollmien-Schlichting wavepackets located far downstream in streamwise direction, whereas the leading optimal disturbances are tilted structures located far upstream in the boundary layer. For control design on the other hand, the input-output behavior of the system is of interest and the snapshot-method is employed to compute balanced modes that correctly capture this behavior. The inputs are external disturbances and wall actuation and the outputs are sensors that extract wall shear stress. A low-dimensional model that capture the input-output behavior is constructed by projection onto balanced modes. The reduced-order model is then used to design a feedback control strategy such that the growth of disturbances are damped as they propagate downstream.

  • 2.
    Bagheri, Shervin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Brandt, Luca
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Henningson, Dan Stefan
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Input-output analysis and control design of spatially developing shear flows2008Ingår i: 5th AIAA Theoretical Fluid Mechanics Conference, 2008Konferensbidrag (Refereegranskat)
    Abstract [en]

    A framework for the input-output analysis, model reduction and control design of spatially developing shear flows is presented using the Blasius boundary-layer flow as an example. An input-output formulation of the governing equations yields a flexible formulation for treating stability problems and for developing control strategies that optimize given objectives. Model reduction plays an important role in this process since the dynamical systems that describe most flows are discretized partial differential equations with a very large number of degrees of freedom. Moreover, as system theoretical tools, such as controllability, observability and balancing has become computationally tractable for large-scale systems, a systematic approach to model reduction is presented.

  • 3.
    Chevalier, Mattias
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Hoepffner, Jérôme
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Linear feedback control and estimation applied to instabilities in spatially developing boundary layers2007Ingår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 588, s. 163-187Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper presents the application of feedback control to spatially developing boundary layers. It is the natural follow-up of Hogberg & Henningson (J. Fluid Mech. vol. 470, 2002, p. 151), where exact knowledge of the entire flow state was assumed for the control. We apply recent developments in stochastic models for the external sources of disturbances that allow the efficient use of several wall measurements for estimation of the flow evolution: the two components of the skin friction and the pressure fluctuation at the wall. Perturbations to base flow profiles of the family of Falkner-Skan-Cooke boundary layers are estimated by use of wall measurements. The estimated state is in turn fed back for control in order to reduce the kinetic energy of the perturbations. The control actuation is achieved by means of unsteady blowing and suction at the wall. Flow perturbations are generated in the upstream region in the computational box and propagate in the boundary layer. Measurements are extracted downstream over a thin strip, followed by a second thin strip where the actuation is performed. It is shown that flow disturbances can be efficiently estimated and controlled in spatially evolving boundary layers for a wide range of base flows and disturbances.

  • 4.
    Chevalier, Mattias P.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Hœpffner, JérÔme
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Feedback control in spatially growing boundary layers2006Konferensbidrag (Refereegranskat)
    Abstract [en]

    Linear feedback control has been applied to transitional boundary layer flows. Information from wall-mounted sensors is used to estimate the flow state. The estimated state is then used to compute the optimal feedback control which is applied as blowing and suction with zero net mass-flux through the wall. The performance of the controller is tested in direct numerical simulations of a spatially growing Falkner-Skan- Cooke boundary layer where an inflectional instability is triggered. The extension to spatial boundary layer flows is an important step towards real applications.

  • 5.
    Kierkegaard, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Farkost och flyg, MWL Strömningsakustik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Efraimsson, Gunilla
    KTH, Skolan för teknikvetenskap (SCI), Farkost och flyg. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Heopffner, Jerome
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Henningson, Dan
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Åbom, Mats
    KTH, Skolan för teknikvetenskap (SCI), Farkost och flyg, MWL Strömningsakustik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Identifications of sources of sound in low Mach number flows by the use of flow field eigenmodes2006Ingår i: 13th International Congress on Sound and Vibration 2006, 2006, s. 2967-2974Konferensbidrag (Refereegranskat)
    Abstract [en]

    We present a method to study sound generation processes in low Mach Number flows. Instead of the full flow field obtained from e.g a DNS, we consider a base flow together with a time-dependent perturbation ,where the perturbation satisfy the Navier-Stokes equations linearized around the base-flow. In a reduced model the perturbation is approximated by a linear combination of the cigenmodes of a corresponding eigenvalue problem. The behavior in time is determined by the corresponding eigenvalues. Curie's equation is used to calculate the acoustic field. By studying the source terms in Curie's equation, it is possible to identify mechanisms for sources of sound. This makes it possible to study how the different sources of sound depend on different structures of the flow field. We apply the methodology on a two- dimensional flow over a cavity with smoothed corners. Results of acoustic pressure in the far field and source strengths for different superpositions of eigenmodes are presented.

  • 6.
    Kierkegaard, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Farkost och flyg, MWL Marcus Wallenberg Laboratoriet.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Efraimsson, Gunilla
    KTH, Skolan för teknikvetenskap (SCI), Farkost och flyg, Aeroakustik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Henningson, Dan Stefan
    KTH, Skolan för teknikvetenskap (SCI), Mekanik, Stabilitet, Transition, Kontroll.
    Flow field eigenmode decompositions in aeroacoustics2010Ingår i: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 39, nr 2, s. 338-344Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper an efficient method to study sound generation processes in low Mach number flows is presented. We apply the methodology on a two-dimensional flow over a cavity with smoothed corners. Instead of the full flow field obtained from, for example a Direct Numerical Simulation (DNS), we use a reduced model based on global modes to obtain the aeroacoustic sources. Global modes are eigenmodes to the Navier-Stokes equations, linearized about a steady base flow. In a reduced model the perturbations from a steady state are approximated by a linear combination of the eigenmodes. The time dependence is determined by the corresponding eigenvalues. Curie's equation is used to calculate the acoustic field, and by studying the source terms in Curie's equation, mechanisms for sources of sound are identified. Results of acoustic pressure in the far-field and source strengths for different superpositions of eigenmodes are presented.

  • 7.
    Monokrousos, Antonios
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Brandt, Luca
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Global optimal disturbances in the Blasius flow using time-steppersManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The stability of the Blasius flat-plate boundary-layer flow to three-dimensional disturbances is studied by means of optimisation methods at relatively high Reynolds numbers. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. Both optimisation problems are solved using a Lagrange multiplier technique, where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearised Navier-Stokes equations. In both cases the evolution equations for the Lagrange multiplier are the adjoint Navier-Stokes equations. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. The optimal initial condition for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with oblique wave packet propagation that dominates. It is found that the latter mechanism is dominant for the relatively high Reynolds number and the long computational domain considered here. The spatial structure of the optimal forcing is similar to the that of the optimal initial condition, and the response to forcing is also dominated by the Orr/oblique wave mechanism, however less so than in the former case. The lift-up mechanism is, as in the local approach using the Orr-Sommerfeld squire equations, most efficient at zero frequency and degrades slowly for increasing frequencies.

  • 8.
    Monokrousos, Antonios
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Brandt, Luca
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers2010Ingår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 650, s. 181-214Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The global linear stability of the flat-plate boundary-layer flow to three-dimensional disturbances is studied by means of an optimization technique. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. Both optimization problems are solved using a Lagrange multiplier technique, where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearized Navier-Stokes equations. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. In addition, the use of matrix-free methods enables us to extend the present framework to any geometrical configuration. The optimal initial condition for spanwise wavelengths of the order of the boundary-layer thickness are finite-length streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths, it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. This mechanism is dominant for the long computational domain and thus for the relatively high Reynolds number considered here. Three-dimensional localized optimal initial conditions are also computed and the corresponding wave packets examined. For short optimization times, the optimal disturbances consist of streaky structures propagating and elongating in the downstream direction without significant spreading in the lateral direction. For long optimization times, we find the optimal disturbances with the largest energy amplification. These are wave packets of Tollmien-Schlichting waves with low streamwise propagation speed and faster spreading in the spanwise direction. The pseudo-spectrum of the system for real frequencies is also computed with matrix-free methods. The spatial structure of the optimal forcing is similar to that of the optimal initial condition, and the largest response to forcing is also associated with the Orr/oblique wave mechanism, however less so than in the case of the optimal initial condition. The lift-up mechanism is most efficient at zero frequency and degrades slowly for increasing frequencies. The response to localized upstream forcing is also discussed.

  • 9.
    Monokrousos, Antonios
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW. KTH, Skolan för teknikvetenskap (SCI), Mekanik, Stabilitet, Transition, Kontroll.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW. KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Brandt, Luca
    KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW. KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW. KTH, Skolan för teknikvetenskap (SCI), Mekanik, Stabilitet, Transition, Kontroll.
    Optimal disturbances with iterative methods2010Ingår i: SEVENTH IUTAM SYMPOSIUM ON LAMINAR-TURBULENT TRANSITION / [ed] Schlatter P; Henningson DS, 2010, Vol. 18, s. 533-536Konferensbidrag (Refereegranskat)
  • 10.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Feedback Control of Spatially Evolving Flows2007Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    In this thesis we apply linear feedback control to spatially evolving flows in order to minimize disturbance growth. The dynamics is assumed to be described by the linearized Navier--Stokes equations. Actuators and sensor are designed and a Kalman filtering technique is used to reconstruct the unknown flow state from noisy measurements. This reconstructed flow state is used to determine the control feedback which is applied to the Navier--Stokes equations through properly designed actuators. Since the control and estimation gains are obtained through an optimization process, and the Navier--Stokes equations typically forms a very high-dimensional system when discretized there is an interest in reducing the complexity of the equations. One possible approach is to perform Fourier decomposition along (almost) homogeneous spatial directions and another is by constructing a reduced order model by Galerkin projection on a suitable set of vectors. The first strategy is used to control the evolution of a range of instabilities in the classical family of Falkner--Skan--Cooke flows whereas the second is applied to a more complex cavity type of geometry.

  • 11.
    Åkervik, Espen
    KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW. KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Global stability and feedback control of boundary layer flows2008Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    In this thesis the stability of generic boundary layer flows is studied from a global viewpoint using optimization methods. Global eigenmodes of the incompressible linearized Navier-Stokes equations are computed using the Krylov subspace Arnoldi method. These modes serve as a tool both to study asymptotic stability and as a reduced basis to study transient growth. Transient growth is also studied using adjoint iterations. The knowledge obtained from the stability analysis is used to device systematic feedback control in the Linear Quadratic Gaussian framework. The dynamics is assumed to be described by the linearized Navier-Stokes equations. Actuators and sensors are designed and a Kalman filtering technique is used to reconstruct the unknown flow state from noisy measurements. This reconstructed flow state is used to determine the control feedback which is applied to the Navier-Stokes equations through properly designed actuators. Since the control and estimation gains are obtained through an optimization process, and the Navier-Stokes equations typically forms a very high-dimensional system when discretized there is an interest in reducing the complexity of the equations. A standard method to construct a reduced order model is to perform a Galerkin projection of the full equations onto the subspace spanned by a suitable set of vectors, such as global eigenmodes and balanced truncation modes.

  • 12.
    Åkervik, Espen
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Brandt, Luca
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Hoepffner, Jérôme
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Marxen, Olaf
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Schlatter, Philipp
    KTH, Skolan för teknikvetenskap (SCI), Mekanik.
    Steady solutions of the Navier-Stokes equations by selective frequency damping2006Ingår i: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 18, nr 6, s. 068102-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A new method, enabling the computation of steady solutions of the Navier-Stokes equations in globally unstable configurations, is presented. We show that it is possible to reach a steady state by damping the unstable (temporal) frequencies. This is achieved by adding a dissipative relaxation term proportional to the high-frequency content of the velocity fluctuations. Results are presented for cavity-driven boundary-layer separation and a separation bubble induced by an external pressure gradient.

  • 13.
    Åkervik, Espen
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Ehrenstein, Uwe
    IRPHÉ, Université de Provence.
    Gallaire, Francois
    Laboratoire J.A. Dieudonné.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Global two-dimensional stability measures of the flat plate boundary-layer flow2008Ingår i: European journal of mechanics. B, Fluids, ISSN 0997-7546, E-ISSN 1873-7390, Vol. 27, nr 5, s. 501-513Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The stability of the two-dimensional flat plate boundary-layer is studied by means of global eigenmodes. These eigenmodes depend both on the streamwise and wall-normal coordinate, hence there are no assumptions on the streamwise length scales of the disturbances. Expanding the perturbation velocity field in the basis of eigenmodes yields a reduced order model from which the stability characteristics of the flow, i.e. the initial condition and forcing function leading to the largest energy growth, are extracted by means of non-modal analysis. In this paper we show that, even when performing stability analysis using global eigenmodes, it is not sufficient to consider only a few of the least damped seemingly relevant eigenmodes. Instead it is the task of the optimization procedure, inherent in the non-modal analysis, to decide which eigenmodes are relevant. We show that both the optimal initial condition and the optimal forcing structure have the form of upstream tilted structures. Time integration reveals that these structures gain energy through the so called Orr mechanism, where the instabilities extract energy from the mean shear. This provides the optimal way of initiating Tollmien-Schlichting waves in the boundary layer. The optimal initial condition results in a localized Tollmien-Schlichting wavepacket that propagates downstream, whereas the optimal forcing results in a persistent Tollmien-Schlichting wave train.

  • 14.
    Åkervik, Espen
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Haepffner, Jerome
    Ehrenstein, Uwe
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Model reduction and control of a cavity-driven separated boundary layer2008Ingår i: IUTAM Symposium On Flow Control And Mems / [ed] Morrison, JF; Birch, DM; Lavoie, P, 2008, Vol. 7, s. 147-155Konferensbidrag (Refereegranskat)
    Abstract [en]

    The control of a globally unstable boundary-layer flow along a two-dimensional cavity is considered. When perturbed by the worst-case initial condition, the flow exhibits a large transient growth associated with the development of a wave packet along the cavity shear layer followed by a global cycle related to the least stable global eigenmodes. The flow simulation procedure is coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced model for the control optimization is obtained by a projection on the least stable global eigenmodes. The LQG controller is run in parallel to the Navier-Stokes time integration. It is shown that the controller is able to damp out the global oscillations.

  • 15.
    Åkervik, Espen
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Hoepffner, Jérôme
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Uwe, Eherenstein
    IRPH́E, Université de Provence.
    Henningson, Dan S.
    KTH, Skolan för teknikvetenskap (SCI), Mekanik. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
    Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes2007Ingår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 579, s. 305-314Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Two-dimensional global eigenmodes are used as a projection basis both for analysing the dynamics and building a reduced model for control in a prototype separated boundary-layer flow. In the present configuration, a high aspect ratio smooth cavity-like geometry confines the separation bubble. Optimal growth analysis using the reduced basis shows that the sum of the highly non-normal global eigenmodes are able to describe a localized disturbance. Subject to this worst-case initial condition, a large transient growth associated with the development of a wavepacket along the shear layer followed by a global cycle related to the two unstable global eigenmodes is found. The flow simulation procedure is coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced model for the control optimization is obtained by a projection on the least stable global eigenmodes, and the resulting linear-quadratic-gaussian controller is applied to the Navier--Stokes time integration. It is shown that the controller is able to damp out the global oscillations.

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