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Computing Optimal Forcing Using Laplace Preconditioning
KTH, School of Engineering Sciences (SCI), Mechanics, Stability, Transition and Control.ORCID iD: 0000-0001-9446-7477
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0001-9627-5903
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0001-7864-3071
2017 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 22, no 5, 1508-1532 p.Article in journal (Refereed) Published
Abstract [en]

For problems governed by a non-normal operator, the leading eigenvalue of the operator is of limited interest and a more relevant measure of the stability is obtained by considering the harmonic forcing causing the largest system response. Various methods for determining this so-called optimal forcing exist, but they all suffer from great computational expense and are hence not practical for large-scale problems. In the present paper a new method is presented, which is applicable to problems of arbitrary size. The method does not rely on timestepping, but on the solution of linear systems, in which the inverse Laplacian acts as a preconditioner. By formulating the search for the optimal forcing as an eigenvalue problem based on the resolvent operator, repeated system solves amount to power iterations, in which the dominant eigenvalue is seen to correspond to the energy amplification in a system for a given frequency, and the eigenfunction to the corresponding forcing function. Implementation of the method requires only minor modifications of an existing timestepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss the method, first, in the context of the linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavity flow governed by the Navier-Stokes equations. Most importantly, we demonstrate that for the lid-driven cavity, the optimal forcing can be computed using a factor of up to 500 times fewer operator evaluations than the standard method based on exponential timestepping.

Place, publisher, year, edition, pages
2017. Vol. 22, no 5, 1508-1532 p.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-214476DOI: 10.4208/cicp.OA-2016-0070ISI: 000408436300012OAI: oai:DiVA.org:kth-214476DiVA: diva2:1148490
Note

QC 20171011

Available from: 2017-10-11 Created: 2017-10-11 Last updated: 2017-11-23Bibliographically approved
In thesis
1. Studies on instability and optimal forcing of incompressible flows
Open this publication in new window or tab >>Studies on instability and optimal forcing of incompressible flows
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers the hydrodynamic instability and optimal forcing of a number of incompressible flow cases. In the first part, the instabilities of three problems that are of great interest in energy and aerospace applications are studied, namely a Blasius boundary layer subject to localized wall-suction, a Falkner–Skan–Cooke boundary layer with a localized surface roughness, and a pair of helical vortices. The two boundary layer flows are studied through spectral element simulations and eigenvalue computations, which enable their long-term behavior as well as the mechanisms causing transition to be determined. The emergence of transition in these cases is found to originate from a linear flow instability, but whereas the onset of this instability in the Blasius flow can be associated with a localized region in the vicinity of the suction orifice, the instability in the Falkner–Skan–Cooke flow involves the entire flow field. Due to this difference, the results of the eigenvalue analysis in the former case are found to be robust with respect to numerical parameters and domain size, whereas the results in the latter case exhibit an extreme sensitivity that prevents domain independent critical parameters from being determined. The instability of the two helices is primarily addressed through experiments and analytic theory. It is shown that the well known pairing instability of neighboring vortex filaments is responsible for transition, and careful measurements enable growth rates of the instabilities to be obtained that are in close agreement with theoretical predictions. Using the experimental baseflow data, a successful attempt is subsequently also made to reproduce this experiment numerically.

In the second part of the thesis, a novel method for computing the optimal forcing of a dynamical system is developed. The method is based on an application of the inverse power method preconditioned by the Laplace preconditioner to the direct and adjoint resolvent operators. The method is analyzed for the Ginzburg–Landau equation and afterwards the Navier–Stokes equations, where it is implemented in the spectral element method and validated on the two-dimensional lid-driven cavity flow and the flow around a cylinder.

Place, publisher, year, edition, pages
Stockholm, Sweden: KTH Royal Institute of Technology, 2017. 47 p.
Series
TRITA-MEK, ISSN 0348-467X ; 2017:19
Keyword
hydrodynamic stability, optimal forcing, resolvent operator, Laplace preconditioner, spectral element method, eigenvalue problems, inverse power method, direct numerical simulations, Falkner–Skan–Cooke boundary layer, localized roughness, crossflow vortices, Blasius boundary layer, localized suction, helical vortices, lid-driven cavity, cylinder flow
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-218172 (URN)978-91-7729-622-5 (ISBN)
Public defence
2017-12-14, D3, Lindstedtsvägen 5, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20171124

Available from: 2017-11-24 Created: 2017-11-23 Last updated: 2017-11-27Bibliographically approved

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Brynjell-Rahkola, MattiasSchlatter, PhilippHenningson, Dan S.

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