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Spectral analysis of nonlinear flows
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0002-8209-1449
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0001-9627-5903
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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.

Place, publisher, year, edition, pages
2009. Vol. 641, 115-127 p.
Keyword [en]
Cross-Flow, Model-Reduction, Systems, Jets, Wake
National Category
Engineering and Technology
URN: urn:nbn:se:kth:diva-11949DOI: 10.1017/S0022112009992059ISI: 000273588900005ScopusID: 2-s2.0-76349094911OAI: diva2:290859
QC 20111004Available from: 2010-01-28 Created: 2010-01-28 Last updated: 2011-10-04Bibliographically approved
In thesis
1. Analysis and control of transitional shear flows using global modes
Open this publication in new window or tab >>Analysis and control of transitional shear flows using global modes
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.
Trita-MEK, ISSN 0348-467X ; 2010:01
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
urn:nbn:se:kth:diva-11894 (URN)978-91-7415-540-2 (ISBN)
Public defence
2010-02-12, F3, Lindsedsv, 26, KTH, Stockholm, 10:15 (English)
Available from: 2010-01-26 Created: 2010-01-20 Last updated: 2010-11-03

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