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Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

Sharma, Ati S. and Mezić, Igor and McKeon, Beverley J. (2016) Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations. Physical Review Fluids, 1 (3). Art. No. 032402(R). ISSN 2469-990X. https://resolver.caltech.edu/CaltechAUTHORS:20160729-123350653

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Abstract

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1103/PhysRevFluids.1.032402DOIArticle
http://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.1.032402PublisherArticle
https://arxiv.org/abs/1603.08378arXivDiscussion Paper
ORCID:
AuthorORCID
Sharma, Ati S.0000-0002-7170-1627
McKeon, Beverley J.0000-0003-4220-1583
Alternate Title:On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
Additional Information:© 2016 American Physical Society. (Received 29 March 2016; published 18 July 2016) No significant data were generated for the purposes of EPSRC's data policy. I.M. was partially supported by ARO Contract No. W911NF-14-C-0102 and AFOSR Grant No. FA9550-12-1-0230. B.J.M. gratefully acknowledges the support of the ONR under Grant No. N000141310739.
Group:GALCIT
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Army Research Office (ARO)W911NF-14-C-0102
Air Force Office of Scientific Research (AFOSR)FA9550-12-1-0230
Office of Naval Research (ONR)N000141310739
Issue or Number:3
Record Number:CaltechAUTHORS:20160729-123350653
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160729-123350653
Official Citation:Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations Ati S. Sharma, Igor Mezić, and Beverley J. McKeon Phys. Rev. Fluids 1, 032402(R)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69316
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:29 Jul 2016 20:10
Last Modified:03 Oct 2019 10:21

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