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A method for computing optimal forcing of convectively unstable flows 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, Stability, Transition and Control.ORCID iD: 0000-0001-9627-5903
KTH, School of Engineering Sciences (SCI), Mechanics, Stability, Transition and Control.ORCID iD: 0000-0001-7864-3071
(English)Manuscript (preprint) (Other academic)
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 problem of finding 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 optimal forcing function. Implementation of the method requires only minor modifications of an existing time-stepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss it in the context of the linear Ginzburg-Landau equation.

National Category
Fluid Mechanics and Acoustics Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-175352OAI: oai:DiVA.org:kth-175352DiVA: diva2:860468
Note

QS 2015

Available from: 2015-10-12 Created: 2015-10-12 Last updated: 2015-10-15Bibliographically approved
In thesis
1. Global stability analysis of three-dimensional boundary layer flows
Open this publication in new window or tab >>Global stability analysis of three-dimensional boundary layer flows
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers the stability and transition of incompressible boundary layers. In particular, the Falkner–Skan–Cooke boundary layer subject to a cylindrical surface roughness, and the Blasius boundary layer with applied localized suction are investigated. These flows are of great importance within the aviation industry, feature complex transition scenarios, and are strongly three-dimensional in nature. Consequently, no assumptions regarding homogeneity in any of the spatial directions are possible, and the stability of the flow is governed by an extensive three-dimensional eigenvalue problem.

The stability of these flows is addressed by high-order direct numerical simulations using the spectral element method, in combination with a Krylov subspace projection method. Such techniques target the long-term behavior of the flow and can provide lower limits beyond which transition is unavoidable. The origin of the instabilities, as well as the mechanisms leading to transition in the aforementioned cases are studied and the findings are reported.

Additionally, a novel method for computing the optimal forcing of a dynamical system is developed. This type of analysis provides valuable information about the frequencies and structures that cause the largest energy amplification in the system. The method is based on the inverse power method, and is discussed in the context of the one-dimensional Ginzburg–Landau equation and a two-dimensional flow case governed by the Navier–Stokes equations.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2015. x, 30 p.
Series
TRITA-MEK, ISSN 0348-467X ; 2015:07
Keyword
Hydrodynamic stability, transition to turbulence, global analysis, boundary layers, roughness, laminar flow control, Stokes/Laplace preconditioner, optimal forcing, crossflow vortices, Ginzburg-Landau, Falkner-Skan-Cooke, Blasius, lid-driven cavity
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-175353 (URN)978-91-7595-725-8 (ISBN)
Presentation
2015-10-30, D3, Lindstedtsvägen 5, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20151015

Available from: 2015-10-15 Created: 2015-10-12 Last updated: 2015-10-15Bibliographically approved

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Brynjell-Rahkola, MattiasSchlatter, PhilippHenningson, Dan S.

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