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Global stability and feedback control of boundary layer flows
KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
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.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , viii, 58 p.
Series
Trita-MEK, ISSN 0348-467X ; 2008:09
Keyword [en]
Stability, Global Stability, Feedback Control, Control, Estimation, Absolute/Convective Instabilities, Model Reduction
National Category
Fluid Mechanics and Acoustics
Identifiers
URN: urn:nbn:se:kth:diva-9547ISBN: 978-91-7415-176-3 (print)OAI: oai:DiVA.org:kth-9547DiVA: diva2:117432
Public defence
2008-12-05, E1, Lindstedtsvägen 3, Stockholm, 10:30 (English)
Opponent
Supervisors
Note
QC 20100924Available from: 2008-11-13 Created: 2008-11-12 Last updated: 2010-09-24Bibliographically approved
List of papers
1. Global two-dimensional stability measures of the flat plate boundary-layer flow
Open this publication in new window or tab >>Global two-dimensional stability measures of the flat plate boundary-layer flow
2008 (English)In: European journal of mechanics. B, Fluids, ISSN 0997-7546, E-ISSN 1873-7390, Vol. 27, no 5, 501-513 p.Article in journal (Refereed) Published
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.

Keyword
boundary layer stability, global modes, convective instabilities, non-modal stability
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-9542 (URN)10.1016/j.euromechflu.2007.09.004 (DOI)000258752500001 ()2-s2.0-47549119097 (Scopus ID)
Note
QC 20100923Available from: 2008-11-12 Created: 2008-11-12 Last updated: 2010-11-22Bibliographically approved
2. Steady solutions of the Navier-Stokes equations by selective frequency damping
Open this publication in new window or tab >>Steady solutions of the Navier-Stokes equations by selective frequency damping
Show others...
2006 (English)In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 18, no 6, 068102- p.Article in journal (Refereed) Published
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.

Keyword
approximate deconvolution model, flows, simulation
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-6804 (URN)10.1063/1.2211705 (DOI)000238731500045 ()2-s2.0-33745610422 (Scopus ID)
Note
QC 20100830Available from: 2008-11-13 Created: 2008-11-13 Last updated: 2011-12-16Bibliographically approved
3. Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes
Open this publication in new window or tab >>Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes
2007 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 579, 305-314 p.Article in journal (Refereed) Published
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.

Keyword
Boundary layer flow, Eigenvalues and eigenfunctions, Feedback control, Flow control, Mathematical models, Optimization, Shear stress, Control optimization, Flow simulation, Global eigenmodes
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-9543 (URN)10.1017/S0022112007005496 (DOI)000247023600011 ()2-s2.0-34548175491 (Scopus ID)
Note
QC 20100923Available from: 2008-11-13 Created: 2008-11-12 Last updated: 2010-09-23Bibliographically approved
4. Matrix-free methods for the stability and control of boundary layers
Open this publication in new window or tab >>Matrix-free methods for the stability and control of boundary layers
2009 (English)In: AIAA Journal, ISSN 0001-1452, E-ISSN 1533-385X, Vol. 47, no 5, 1057-1068 p.Article in journal (Refereed) Published
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.

Keyword
Adjoint-based optimization, Blasius boundary layer, Control design, Critical steps, Eigen modes, Eigenvalue problem, Energy growth, External disturbances, Feedback control strategies, Feedback controller, Finite time intervals, High-dimensional systems, Initial conditions, Input-output behavior, Linearized navier-stokes equations, Low-dimensional models, matrix, Optimal disturbances, Reduced order models, Snapshot method, Space and time, Stability analysis, Stability and control, Stability properties, Streamwise directions, Tollmien-schlichting wave packets, Velocity field, Wall shear stress, Aerodynamics, Boundary layers, Eigenvalues and eigenfunctions, Feedback control, Flow separation, Navier Stokes equations, Optimization, System stability
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-9546 (URN)10.2514/1.41365 (DOI)000265586200001 ()2-s2.0-67649170531 (Scopus ID)
Note
QC 20100927 AIAA 5th Theoretical Fluid Mechanics Meeting, Seattle, WA, JUN 23-26, 2008Available from: 2008-11-12 Created: 2008-11-12 Last updated: 2011-03-29Bibliographically approved
5. Linear feedback control and estimation applied to instabilities in spatially developing boundary layers
Open this publication in new window or tab >>Linear feedback control and estimation applied to instabilities in spatially developing boundary layers
2007 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 588, 163-187 p.Article in journal (Refereed) Published
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.

Keyword
optimal perturbations, state estimation, flow systems, transition, wall, stability, equations
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-6803 (URN)10.1017/S0022112007007392 (DOI)000250675400008 ()2-s2.0-37749017590 (Scopus ID)
Note
QC 20100830Available from: 2007-02-22 Created: 2007-02-22 Last updated: 2010-09-23Bibliographically approved
6. Global optimal disturbances in the Blasius flow using time-steppers
Open this publication in new window or tab >>Global optimal disturbances in the Blasius flow using time-steppers
(English)Manuscript (preprint) (Other academic)
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.

National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-9545 (URN)
Note
QC 20100924Available from: 2008-12-03 Created: 2008-11-12 Last updated: 2010-09-24Bibliographically approved

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  • harvard1
  • ieee
  • modern-language-association-8th-edition
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  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
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Output format
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