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Analysis and control of transitional shear flows using global modes
KTH, School of Engineering Sciences (SCI), Mechanics.ORCID iD: 0000-0002-8209-1449
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis direct numerical simulations are used to investigate two phenomenain shear flows: laminar-turbulent transition over a flat plate and periodicvortex shedding induced by a jet in cross flow. The emphasis is on understanding and controlling the flow dynamics using tools from dynamical systems and control theory. In particular, the global behavior of complex flows is describedand low-dimensional models suitable for control design are developed; this isdone by decomposing the flow into global modes determined from spectral analysisof various linear operators associated with the Navier–Stokes equations.Two distinct self-sustained global oscillations, associated with the sheddingof vortices, are identified from direct numerical simulations of the jet incrossflow. The investigation is split into a linear stability analysis of the steadyflow and a nonlinear analysis of the unsteady flow. The eigenmodes of theNavier–Stokes equations, linearized about an unstable steady solution revealthe presence of elliptic, Kelvin-Helmholtz and von K´arm´an type instabilities.The unsteady nonlinear dynamics is decomposed into a sequence of Koopmanmodes, determined from the spectral analysis of the Koopman operator. Thesemodes represent spatial structures with periodic behavior in time. A shearlayermode and a wall mode are identified, corresponding to high-frequency andlow-frequency self-sustained oscillations in the jet in crossflow, respectively.The knowledge of global modes is also useful for transition control, wherethe objective is to reduce the growth of small-amplitude disturbances to delaythe transition to turbulence. Using a particular basis of global modes, knownas balanced modes, low-dimensional models that capture the behavior betweenactuator and sensor signals in a flat-plate boundary layer are constructed andused to design optimal feedback controllers. It is shown that by using controltheory in combination with sensing/actuation in small, localized, regionsnear the rigid wall, the energy of disturbances may be reduced by an order of magnitude.

Place, publisher, year, edition, pages
Stockholm: KTH , 2010. , viii, 82 p.
Series
Trita-MEK, ISSN 0348-467X ; 2010:01
Keyword [en]
Fluid mechanics, flow control, hydrodynamic stability, global modes, jet in crossflow, flat-plate boundary layer, laminar-turbulent transition, Arnoldi method, Koopman modes, balanced truncation, direct numerical simulations.
National Category
Fluid Mechanics and Acoustics
Identifiers
URN: urn:nbn:se:kth:diva-11894ISBN: 978-91-7415-540-2 (print)OAI: oai:DiVA.org:kth-11894DiVA: diva2:288180
Public defence
2010-02-12, F3, Lindsedsv, 26, KTH, Stockholm, 10:15 (English)
Opponent
Supervisors
Available from: 2010-01-26 Created: 2010-01-20 Last updated: 2010-11-03
List of papers
1. Input-Output Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows
Open this publication in new window or tab >>Input-Output Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows
2009 (English)In: Applied Mechanics Review, ISSN 0003-6900, E-ISSN 1088-8535, Vol. 62, no 2Article in journal (Refereed) Published
Abstract [en]

This review presents a framework for the input-output analysis, model reduction, and control design for fluid dynamical systems using examples applied to the linear complex Ginzburg-Landau equation. Major advances in hydrodynamics stability, such as global modes in spatially inhomogeneous systems and transient growth of non-normal systems, are reviewed. Input-output analysis generalizes hydrodynamic stability analysis by considering a finite-time horizon over which energy amplification, driven by a specific input (disturbances/actuator) and measured at a specific output (sensor), is observed. In the control design the loop is closed between the output and the input through a feedback gain. Model reduction approximates the system with a low-order model, making modern control design computationally tractable for systems of large dimensions. Methods from control theory are reviewed and applied to the Ginzburg-Landau equation in a manner that is readily generalized to fluid mechanics problems, thus giving a fluid mechanics audience an accessible introduction to the subject.

Keyword
control system synthesis, feedback, flow control, flow instability, flow simulation, Ginzburg-Landau theory, hydrodynamics, reduced order systems
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-8499 (URN)10.1115/1.3077635 (DOI)000263717800003 ()2-s2.0-77950642067 (Scopus ID)
Note
QC 20101103Available from: 2008-05-23 Created: 2008-05-23 Last updated: 2017-12-14Bibliographically approved
2. Input-output analysis, model reduction and control of the flat-plate boundary layer
Open this publication in new window or tab >>Input-output analysis, model reduction and control of the flat-plate boundary layer
2009 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 620, 263-298 p.Article in journal (Refereed) Published
Abstract [en]

The dynamics and control of two-dimensional disturbances in the spatially evolving boundary layer oil a flat plate are investigated from an input output viewpoint. A set-up of spatially localized inputs (external disturbances and actuators) and Outputs (objective functions and sensors) is introduced for the control design of convectively unstable flow configurations. From the linearized Navier Stokes equations with the inputs and outputs, controllable, observable and balanced modes are extracted using the snapshot method. A balanced reduced-order model (ROM) is constructed and shown to capture the input output behaviour of the linearized Navier Stokes equations. This model is finally used to design H-2-feedback controller to suppress the growth or two-dimensional perturbations inside the boundary layer.

Keyword
Control design, Dynamics and control, External disturbances, Feedback controller, Flat plate, Input output analysis, Input-output, Linearized navier-stokes equations, Model reduction, Objective functions, Reduced order models, Snapshot method, Two-dimensional perturbation, Unstable flows, Aerodynamics, Boundary layers, Fluid dynamics, Linearization, Two dimensional, Viscous flow
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-8500 (URN)10.1017/S0022112008004394 (DOI)000263384700012 ()2-s2.0-65649154530 (Scopus ID)
Note
QC 20101103 Submitted 2008Available from: 2008-05-23 Created: 2008-05-23 Last updated: 2017-12-14Bibliographically approved
3. 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: 2017-12-14Bibliographically approved
4. Global stability of a jet in crossflow
Open this publication in new window or tab >>Global stability of a jet in crossflow
2009 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 624, 33-44 p.Article in journal (Refereed) Published
Abstract [en]

A linear stability analysis shows that the jet in crossflow is characterized by self-sustained global oscillations for a jet-to-crossflow velocity ratio of 3. A fully three-dimensional unstable steady-state solution and its associated global eigenmodes are computed by direct numerical simulations and iterative eigenvalue routines. The steady flow, obtained by means of selective frequency damping, consists mainly of a (steady) counter-rotating vortex pair (CVP) in the far field and horseshoe-shaped vortices close to the wall. High-frequency unstable global eigenmodes associated with shear-layer instabilities on the CVP and low-frequency modes associated with shedding vortices in the wake of the jet are identified. Furthermore, different spanwise symmetries of the global modes are discussed. This work constitutes the first simulation-based global stability analysis of a fully three-dimensional base flow.

Keyword
Baseflows, Counter-rotating Vortex Pair, Crossflow velocities, Eigen modes, Eigen-value, Far field, Global stability, Global stability analysis, High frequency HF, Jet in crossflow, Low-frequency modes, Shedding vortex, Simulation-based, Steady state solution, Direct numerical simulation, Flow separation, Flow simulation, Groundwater flow, Jets, Linear stability analysis, Three dimensional
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-8501 (URN)10.1017/S0022112009006053 (DOI)000265754000003 ()2-s2.0-65649120402 (Scopus ID)
Note
QC 20101103Available from: 2008-05-23 Created: 2008-05-23 Last updated: 2017-12-14Bibliographically approved
5. Spectral analysis of nonlinear flows
Open this publication in new window or tab >>Spectral analysis of nonlinear flows
Show others...
2009 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 641, 115-127 p.Article in journal (Refereed) Published
Abstract [en]

We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

Keyword
Cross-Flow, Model-Reduction, Systems, Jets, Wake
National Category
Engineering and Technology
Identifiers
urn:nbn:se:kth:diva-11949 (URN)10.1017/S0022112009992059 (DOI)000273588900005 ()2-s2.0-76349094911 (Scopus ID)
Note
QC 20111004Available from: 2010-01-28 Created: 2010-01-28 Last updated: 2017-12-12Bibliographically approved
6. The stabilizing effect of streaks on Tollmien-Schlichting and oblique waves: A parametric study
Open this publication in new window or tab >>The stabilizing effect of streaks on Tollmien-Schlichting and oblique waves: A parametric study
2007 (English)In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 19, no 7, 078103-1-078103-4 p.Article in journal (Refereed) Published
Abstract [en]

The stabilizing effect of finite amplitude streaks on the linear growth of unstable perturbations [Tollmien-Schlichting (TS) and oblique waves] is numerically investigated by means of the nonlinear parabolized stability equations. We have found that for stabilization of a TS-wave, there exists an optimal spanwise spacing of the streaks. These streaks reach their maximum amplitudes close to the first neutral point of the TS-wave and induce the largest distortion of the mean flow in the unstable region of the TS-wave. For such a distribution, the required streak amplitude for complete stabilization of a given TS-wave is considerably lower than for beta=0.45, which is the optimal for streak growth and used in previous studies. We have also observed a damping effect of streaks on the growth rate of oblique waves in Blasius boundary layer and for TS-waves in Falkner-Skan boundary layers.

Keyword
blasius boundary-layer, transition
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-16836 (URN)10.1063/1.2746047 (DOI)000248486100047 ()2-s2.0-34547814063 (Scopus ID)
Note
QC 20100525 Tidigare titel: The stabilizing effect of streaks on TS-waves: A parametric studyAvailable from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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